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Rafael D. Benguria

Helmut Linde

Departmento de Fısica, Pontificia Universidad Catolica de Chile,Casilla 306, Santiago 22, Chile

E-mail address: [email protected]

Departmento de Fısica, Pontificia Universidad Catolica de Chile,Casilla 306, Santiago 22, Chile

Current address: Global Management Office, SAP SI AG, Albert Einstein Allee3, D–64625 Bensheim, Germany

E-mail address: [email protected]

Key words and phrases. amsbook, AMS-LATEX

The work of RB was partially supported by Fondecyt (CHILE) project 106–0651,and CONICYT/PBCT Proyecto Anillo de Investigacion en Ciencia y TecnologıaACT30/2006; the work of HL was partially supported by a doctoral fellowship

from CONICYT (CHILE).

Contents

Part 1. Isoperimetric Inequalities for Eigenvalues of theLaplace Operator 1

Chapter 1. Introduction 3

Chapter 2. Can one hear the shape of a drum? 52.1. Introduction 52.2. One cannot hear the shape of a drum 102.3. Bibliographical Remarks 13

Chapter 3. Rearrangements 153.1. Definition and basic properties 153.2. Main theorems 163.3. Gradient estimates 183.4. Bibliographical Remarks 20

Chapter 4. The Rayleigh–Faber–Krahn inequality 234.1. The Euclidean case 234.2. Schrodinger operators 244.3. Gaussian Space 254.4. Spaces of constant curvature 264.5. Robin Boundary Conditions 274.6. Bibliographical Remarks 27

Chapter 5. The Szego–Weinberger inequality 315.1. Bibliographical Remarks 33

Chapter 6. The Payne–Polya–Weinberger inequality 356.1. Introduction 356.2. Proof of the Payne–Polya–Weinberger inequality 366.3. Monotonicity of B and g 396.4. The Chiti comparison result 416.5. Schrodinger operators 436.6. Gaussian space 446.7. Spaces of constant curvature 44

Chapter 7. Appendix 477.1. The layer-cake formula 477.2. A consequence of the Brouwer fixed-point theorem 47

Bibliography 49

v

Part 1

Isoperimetric Inequalities forEigenvalues of theLaplace Operator

CHAPTER 1

Introduction

The contents of this manuscript are based on a series of lectures that one ofus (RB) gave in the IV Escuela de Verano en Analisis y Fısica Matematica. TheSummer School took place at the Unidad Cuernavaca del Instituto de Matematicasde la Universidad Nacional Autonoma de Mexico. It is a pleasure to thank theorganizers of the Summer School for their kind invitation and hospitality. Prelimi-nary versions of these lectures were also given in the Short Course in IsoperimetricInequalities for Eigenvalues of the Laplacian, given by one of us (RB) in February of2004, as part of the Thematic Program on Partial Differential Equations held at theFields Institute, in Toronto, and also as part of the course Autovalores del Lapla-ciano y Geometrıa given at the Department of Mathematics of the Universidad dePernambuco, in Recife, Brazil, in August 2003.

Isoperimetric Inequalities have played an important role in mathematics sincethe times of the Ancient Greece. The first and best known isoperimetric inequalityis the classical isoperimetric inequality

A ≤ L2

4π,

relating the area A enclosed by a planar closed curve of perimeter L (i.e., QueenDido’s problem described in Virgilio’s epic poem “The Aeneid”). After the in-troduction of Calculus in the XVII century, many new isoperimetric inequalitieshave been discovered in mathematics and physics (see, e.g., the review articles[B80, O80, P67, PSz51]). The eigenvalues of the Laplacian are “geometric ob-jects” in the sense they do depend on the geometry of the underlying domain,and to some extent (see Chapter 3) the knowledge of the domain characterizes thegeometry of the domain. Therefore it is natural to pose the problem of findingisoperimetric inequalities for the eigenvalues of the Laplacian. The first one toconsider this possibility was Lord Rayleigh in his monograph The Theory of Sound[R45]. In these lectures we will present some of the problems arising in the study ofisoperimetric inequalities for the Laplacian, some of the tools needed in their proofand many bibliographic discussions about the subject. We start our review withthe classical problem of Mark Kac, Can one hear the shape of a drum. In Chapterthree we review the definitions and basic facts about rearrangements of functions.Chapter 4 is devoted to the Rayleigh–Faber–Krahn inequality. In Chapter 5 wereview the Szego–Weinberger inequality, which is an isoperimetric inequality forthe first nontrivial Neumann eigenvalue of the Laplacian. In Chapter 6 we reviewthe Payne–Polya–Weinberger isoperimetric inequality for the quotient of the firsttwo Dirichlet eigenvalues of the Laplacian, as well as several recent extensions.There are many recent interesting isoperimetric results for the eigenvalues of the

3

4 1. INTRODUCTION

bi–harmonic operator, as well as many open problems in that area, which we haveleft out of this review.

We would like to thank the anonymous referee for many useful corrections andremarks.

CHAPTER 2

Can one hear the shape of a drum?

...but it would baffle the most skillfulmathematician to solve the Inverse Problem,and to find out the shape of a bell by means

of the sounds which is capable of sending out.Sir Arthur Schuster (1882).

2.1. Introduction

In 1965, the Committee on Educational Media of the Mathematical Association ofAmerica produced a film on a mathematical lecture by Mark Kac (1914–1984) withthe title: Can one hear the shape of a drum? One of the purposes of the film wasto inspire undergraduates to follow a career in mathematics. An expanded versionof that lecture was later published [K66]. Consider two different smooth, boundeddomains, say Ω1 and Ω2 in the plane. Let 0 < λ1 < λ2 ≤ λ3 ≤ . . . be the sequenceof eigenvalues of the Laplacian on Ω1, with Dirichlet boundary conditions and,correspondingly, 0 < λ′1 < λ′2 ≤ λ′3 ≤ . . . be the sequence of Dirichlet eigenvaluesfor Ω2. Assume that for each n, λn = λ′n (i.e., both domains are isospectral). Then,Mark Kac posed the following question: Are the domains Ω1 and Ω2 congruent inthe sense of Euclidean geometry?.

In 1910, H. A. Lorentz, at the Wolfskehl lecture at the University of Gottingen,reported on his work with Jeans on the characteristic frequencies of the electro-magnetic field inside a resonant cavity of volume Ω in three dimensions. Accordingto the work of Jeans and Lorentz, the number of eigenvalues of the electromagneticcavity whose numerical values is below λ (this is a function usually denoted byN(λ)) is given asymptotically by

(2.1) N(λ) ≈ |Ω|6π2

λ3/2,

for large values of λ, for many different cavities with simple geometry (e.g., cubes,spheres, cylinders, etc.) Thus, according to the calculations of Jeans and Lorentz,to leading order in λ, the counting function N(λ) seemed to depend only on thevolume of the electromagnetic cavity |Ω|. Apparently David Hilbert (1862–1943),who was attending the lecture, predicted that this conjecture of Lorentz would notbe proved during his lifetime. This time, Hilbert was wrong, since his own student,Hermann Weyl (1885–1955) proved the conjecture less than two years after theLorentz’ lecture.

5

6 2. CAN ONE HEAR THE SHAPE OF A DRUM?

Remark: There is a nice account of the work of Hermann Weyl on the eigenvalues ofa membrane in his 1948 J. W. Gibbs Lecture to the American Mathematical Society[We50].

In N dimensions, (2.1) reads,

(2.2) N(λ) ≈ |Ω|(2π)N

CNλN/2,

where CN = π(N/2)/Γ((N/2)+1)) denotes the volume of the unit ball in N dimen-sions.

Using Tauberian theorems, one can relate the behavior of the counting functionN(λ) for large values of λ with the behavior of the function

(2.3) ZΩ(t) ≡∞∑

n=1

exp−λnt,

for small values of t. The function ZΩ(t) is the trace of the heat kernel for thedomain Ω, i.e., ZΩ(t) = tr exp(∆t). As we mention above, λn(Ω) denotes the nDirichlet eigenvalue of the domain Ω.

An example: the behavior of ZΩ(t) for rectangles

With the help of the Riemann Theta function Θ(x), it is simple to compute the traceof the heat kernel when the domain is a rectangle of sides a and b, and therefore toobtain the leading asymptotic behavior for small values of t. The Riemann Thetafunction is defined by

(2.4) Θ(x) =∞∑

n=−∞e−πn2x,

for x > 0. The function Θ(x) satisfies the following modular relation,

(2.5) Θ(x) =1√x

Θ(1x

).

Remark: This modular form for the Theta Function already appears in the classicalpaper of Riemann [Ri859] (manuscript where Riemann puts forward his famousRiemann Hypothesis). In that manuscript, the modular form is attributed to Jacobi.

The modular form (2.5) may be obtained from a very elegant application ofFourier Analysis (see, e.g., [CH53], pp. 75–76) which we reproduce here for com-pleteness. Define

(2.6) ϕx(y) =∞∑

n=−∞e−π(n+y)2x.

Clearly, Θ(x) = ϕx(0). Moreover, the function ϕx(y) is periodic in y of period 1.Thus, we can express it as follows,

(2.7) ϕx(y) =∞∑

k=−∞ake2πki y,

2.1. INTRODUCTION 7

where, the Fourier coefficients are

(2.8) ak =∫ 1

0

ϕk(y)e−2πki y dy.

Replacing the expression (2.6) for ϕx(y) in (2.9), using the fact that e2πki n = 1,we can write,

(2.9) ak =∫ 1

0

∞∑n=−∞

e−π(n+y)2xe−2πik(y+n) dy.

Interchanging the order between the integral and the sum, we get,

(2.10) ak =∞∑

n=−∞

∫ 1

0

(e−π(n+y)2xe−2πik(y+n)

)dy.

In the nth summand we make the change of variables y → u = n + y. Clearly, uruns from n to n + 1, in the nth summand. Thus, we get,

(2.11) ak =∫ ∞

−∞e−πu2xe−2πik u du.

i.e., ak is the Fourier transform of a Gaussian. Thus, we finally obtain,

(2.12) ak =1√x

e−πk2/x.

Since, Θ(x) = ϕx(0), from (2.7) and (2.12) we finally get,

(2.13) Θ(x) =∞∑

k=−∞ak =

1√x

∞∑

k=−∞e−πk2/x =

1√x

Θ(1x

).

Remarks: i) The method exhibited above is a particular case of the Poisson Sum-mation Formula. See [CH53], pp. 76–77; ii) It should be clear from (2.4) thatlimx→∞Θ(x) = 1. Hence, from the modular form for Θ(x) we immediately seethat

(2.14) limx→0

√xΘ(x) = 1.

Once we have the modular form for the Riemann Theta function, it is simpleto get the leading asymptotic behavior of the trace of the heat kernel ZΩ(t), forsmall values of t, when the domain Ω is a rectangle. Take Ω to be the rectangle ofsides a and b. Its Dirichlet eigenvalues are given by

(2.15) λn,m = π2

[n2

a2+

m2

b2

],

with n,m = 1, 2, . . . . In terms of the Dirichlet eigenvalues, the trace of the heatkernel, ZΩ(t) is given by

(2.16) ZΩ(t) =∞∑

n,m=1

e−λn,mt.


and using (2.15), and the definition of Θ(x), we get,

(2.17) ZΩ(t) =14

[θ(

π t

a2)− 1

] [θ(

π t

b2)− 1

].

Using the asymptotic behavior of the Theta function for small arguments, i.e.,(2.14) above, we have

(2.18) ZΩ(t) ≈ 14(

a√π t

− 1)(b√π t

− 1) ≈ 14πt

ab− 14√

πt(a + b) +

14

+ O(t).

In terms of the area of the rectangle A = ab and its perimeter L = 2(a + b), theexpression ZΩ(t) for the rectangle may be written simply as,

(2.19) ZΩ(t) ≈ 14πt

A− 18√

πtL +

14

+ O(t).

Remark: Using similar techniques, one can compute the small t behavior of ZΩ(t)for various simple regions of the plane (see, e.g., [McH94]).

The fact that the leading behavior of ZΩ(t) for t small, for any bounded, smoothdomain Ω in the plane is given by

(2.20) ZΩ(t) ≈ 14πt

A

was proven by Hermann Weyl [We11]. Here, A = |Ω| denotes the area of Ω. In fact,what Weyl proved in [We11] is the Weyl Asymptotics of the Dirichlet eigenvalues,i.e., for large n, λn ≈ (4π n)/A. Weyl’s result (2.20) implies that one can hear thearea of the drum.

In 1954, the Swedish mathematician, Ake Pleijel [Pj54] obtained the improvedasymptotic formula,

Z(t) ≈ A

4πt− L

8√

πt,

where L is the perimeter of Ω. In other words, one can hear the area and theperimeter of Ω. It follows from Pleijel’s asymptotic result that if all the frequenciesof a drum are equal to those of a circular drum then the drum must itself becircular. This follows from the classical isoperimetric inequality (i.e., L2 ≥ 4πA,with equality if and only if Ω is a circle). In other words, one can hear whether adrum is circular. It turns out that it is enough to hear the first two eigenfrequenciesto determine whether the drum has the circular shape [AB91]

In 1966, Mark Kac obtained the next term in the asymptotic behavior of Z(t)[K66]. For a smooth, bounded, multiply connected domain on the plane (with rholes)

(2.21) Z(t) ≈ A

4πt− L

8√

πt+

16(1− r).

Thus, one can hear the connectivity of a drum. The last term in the above asymp-totic expansion changes for domains with corners (e.g., for a rectangular membrane,using the modular formula for the Theta Function, we obtained 1/4 instead of 1/6).Kac’s formula (2.21) was rigorously justified by McKean and Singer [McKS67].

2.1. INTRODUCTION 9

Moreover, for domains having corners they showed that each corner with inte-rior angle γ makes an additional contribution to the constant term in (2.21) of(π2 − γ2)/(24πγ).

An sketch of Kac’s analysis for the first term of the asymptotic expansion is asfollows (here we follow [K66, McH94]). If we imagine some substance concentratedat ~ρ = (x0, y0) diffusing through the domain Ω bounded by ∂Ω, where the substanceis absorbed at the boundary, then the concentration PΩ(~p

∣∣ ~r; t) of matter at ~r =(x, y) at time t obeys the diffusion equation

∂PΩ

∂t= ∆PΩ

with boundary condition PΩ(~p∣∣ ~r; t) → 0 as ~r → ~a, ~a ∈ ∂Ω, and initial condition

PΩ(~p∣∣ ~r; t) → δ(~r − ~p) as t → 0, where δ(~r − ~p) is the Dirac delta function. The

concentration PΩ(~p∣∣ ~r; t) may be expressed in terms of the Dirichlet eigenvalues of

Ω, λn and the corresponding (normalized) eigenfunctions φn as follows,

PΩ(~p∣∣ ~r; t) =

∞∑n=1

e−λntφn(~p)φn(~r).

For small t, the diffusion is slow, that is, it will not feel the influence of the boundaryin such a short time. We may expect that

PΩ(~p∣∣ ~r; t) ≈ P0(~p

∣∣ ~r; t),

ar t → 0, where ∂P0/∂t = ∆P0, and P0(~p∣∣ ~r; t) → δ(~r − ~p) as t → 0. P0 in fact

represents the heat kernel for the whole R, i.e., no boundaries present. This kernelis explicitly known. In fact,

P0(~p∣∣ ~r; t) =

14πt

exp(−|~r − ~p|2/4t),

where |~r − ~p|2 is just the Euclidean distance between ~p and ~r. Then, as t → 0+,

PΩ(~p∣∣ ~r; t) =

∞∑n=1

e−λntφn(~p)φn(~r) ≈ 14πt

exp(−|~r − ~p|2/4t).

Thus, when set ~p = ~r we get∞∑

n=1

e−λntφ2n(~r) ≈ 1

4πt.

Integrating both sides with respect to ~r, using the fact that φn is normalized, wefinally get,

∞∑n=1

e−λn t ≈ |Ω|4πt

,

which is the first term in the expansion (2.21). Further analysis gives the remainingterms (see [K66]).


Figure 1. GWW Isospectral Domains D1 and D2

2.2. One cannot hear the shape of a drum

In the quoted paper of Mark Kac [K66] he says that he personally believed that onecannot hear the shape of a drum. A couple of years before Mark Kac’ article, JohnMilnor [Mi64], had constructed two non-congruent sixteen dimensional tori whoseLaplace–Beltrami operators have exactly the same eigenvalues. In 1985 ToshikazuSunada [Su85], then at Nagoya University in Japan, developed an algebraic frame-work that provided a new, systematic approach of considering Mark Kac’s question.Using Sunada’s technique several mathematicians constructed isospectral mani-folds (e.g., Gordon and Wilson; Brooks; Buser, etc.). See, e.g., the review article ofRobert Brooks (1988) with the situation on isospectrality up to that date in [Br88].Finally, in 1992, Carolyn Gordon, David Webb and Scott Wolpert [GWW92] gavethe definite negative answer to Mark Kac’s question and constructed two plane do-mains (henceforth called the GWW domains) with the same Dirichlet eigenvalues.

Proof of Isospectrality Using Transplantation:

The most elementary proof of isospectrality of the GWW domains is done usingthe method of transplantation. For the method of transplantation see, e.g., [Be92,Be93]. See also the expository article [Be93b] by the same author. The methodalso appears briefly described in the article of Sridhar and Kudrolli cited in theBibliographical Remarks, iii) at the end of this chapter.

To conclude this chapter we will give the details of the proof of isospectralityof the GWW domains using transplantation. For that purpose label from 1 to 7the congruent triangles that make the two GWW domains (see Figure 1). Each ofthis isosceles right triangles has two cathets, labeled A and B and the hypothenuse,labeled T . Each of the pieces (triangles) that make each one of the two domains isconnected to one or more neighboring triangles through a side A, a side B or a sideT . Each of the two isospectral domains has an associated graph, which are givenin Figure 2.

These graphs have their origin in the algebraic formulation of Sunada [Su85].The vertices in each graph are labeled according to the number that each of thepieces (triangles) has in each of the given domains. As for the edges joining twovertices in these graphs, they are labeled by either an A, a B or a T depending on

2.2. ONE CANNOT HEAR THE SHAPE OF A DRUM 11

Figure 2. Sunada Graphs corresponding to Domains D1 and D2

the type of the common side of two neighboring triangles in Figure 1. In order toshow that both domains are isospectral it is convenient to consider any functiondefined on each domain as consisting of seven parts, each part being the restrictionof the original function to each one of the individual triangles that make the domain.In this way, if ψ is a function defined on the domain 1, we will write as a vector withseven components, i.e., ψ = [ψi]7i=1, where ψi is a scalar function whose supportis triangle i on the domain 1. Similarly, a function ϕ defined over the domain 2may be represented as a seven component vector ϕ = [ϕi]7i=1, with the equivalentmeaning but referred to the second domain.

In order to show the isospectrality of the two domains we have to exhibit a map-ping transforming the functions defined on the first domain into functions definedin the second domain. Given the decomposition we have made of the eigenfunctionsas vectors of seven components, this transformation will be represented by a 7× 7matrix. In order to show that the two domains have the same spectra we need thismatrix to be orthogonal. This matrix is given explicitly by

(2.22) TD =

−a a a −a b −b ba −b −a b −a a −ba −a −b b −b a −a−a b b −a a −b a−b a b −a a −a bb −a −a b −a b −a−b b a −a b −a a

.

For the matrix TD to be orthogonal, we need that the parameters a and b satisfythe following relations: 4a2+3b2 = 1, 2a2+4ab+b2 = 0, and 4a+3b = 1. Althoughwe do not need the numerical values of a and b in the sequel, it is good to knowthat there is a solution to this system of equations, namely a = (1− 3

√8/4)/7 and

b = (1 +√

8)/7. The matrix TD is orthogonal, i.e., TDT tD = 1. The label D used

here refers to the fact that this matrix TD is used to show isospectrality for theDirichlet problem. A similar matrix can be constructed to show isospectrality forthe Neumann problem. In order to show isospectrality it is not sufficient to showthat the matrix TD is orthogonal. It must fulfill two additional properties. On theone hand it should transform a function ψ that satisfies the Dirichlet conditionsin the first domain in a function ϕ that satisfies Dirichlet boundary conditionson the second domain. Moreover, by elliptic regularity, since the functions ψ and


its image ϕ satisfy the eigenvalue equation −∆u = λu, they must be smooth (infact they should be real analytic in the interior of the corresponding domains), andtherefore they must be continuous at the adjacent edges connecting two neighboringtriangles. Thus, while the function ψ is continuous when crossing the common edgesof neighboring triangles in the domain 1, the function ϕ should be continuous whencrossing the common edges of neighboring triangles in the domain 2. These twoproperties are responsible for the peculiar structure of a’s and b’s in the componentsof the matrix TD.

To illustrate these facts, if the function ϕ is an eigenfunction of the Dirichletproblem for the domain 2, it must satisfy, among others, the following properties,

(2.23) ϕA2 = ϕA

7 ,

and

(2.24) ϕT5 = 0.

Here ϕ2 denotes the second component of ϕ, i.e., the restriction of the function ϕ tothe second triangle in Domain 2 (see figure 2). On the other hand, ϕA

2 denotes therestriction of ϕ2 on the edge A of triangle 2. Since on the domain 2, the triangles2 and 7 are glued through a cathet of type A, (2.23) is precisely the condition thatϕ has to be smooth in the interior of 2. On the other hand, ϕ must be a solution ofthe Dirichlet problem for the domain 2 and as such it must satisfy zero boundaryconditions. Since the hypothenuse T of triangle 5 is part of the boundary of thedomain 2, ϕ must vanish there. This is precisely the condition (2.24). Let us check,as an exercise that if ψ is smooth and satisfies Dirichlet boundary conditions inthe domain 1, its image ϕ = TDψ satisfies (2.24) over the domain 2. We let as anexercise to the reader to check (2.23), and all the other conditions on “smoothness”and boundary condition of ϕ (this is a long but straightforward task). From (2.22)we have that

(2.25) ϕT5 = −bψT

1 + aψT2 + bψT

3 − aψT4 + aψT

5 − aψT6 + bψT

7 .

Since all the sides of type T of the pieces 1, 3 and 7 in the domain 1 are part of theboundary of the domain (see figure 1), ψT

1 = ψT3 = ψT

7 = 0. On the other hand,since 2 and 7 are neighboring triangles in the domain 1, glued through a side oftype T , we have ψT

2 = ψT4 . By the same reasoning we have ψT

5 = ψT6 . Using these

three conditions on (2.25) we obtain (2.24). All the other conditions can be verifiedin a similar way. Collecting all these facts, we conclude with

Theorem 2.1 (P. Berard). The transformation TD given by (2.22) is an isom-etry from L2(D1) into L2(D2) (here D1 and D2 are the two domains of Figures 1and 2), which induces an isometry from H1

0 (D1) into H10 (D2).

and therefore we have

Theorem 2.2 (C. Gordon, D. Webb, S. Wolpert). The domains D1 and D2 offigures 1 and 2 are isospectral.

Although the proof by transplantation is straightforward to follow, it does notshed light on the rich geometric, analytic and algebraic structure of the probleminitiated by Mark Kac. For the interested reader it is recommendable to read thepapers of Sunada [Su85] and of Gordon, Webb and Wolpert [GWW92].

2.3. BIBLIOGRAPHICAL REMARKS 13

2.3. Bibliographical Remarks

i) The sentence of Arthur Schuster (1851–1934) quoted at the beginning of this chapteris cited in Reed and Simon’s book, volume IV [RSIV]. It is taken from the article A.Schuster, The Genesis of Spectra, in Report of the fifty–second meeting of the BritishAssociation for the Advancement of Science (held at Southampton in August 1882). Brit.Assoc. Rept., pp. 120–121, 1883. Arthur Schuster was a British physicist (he was a leaderspectroscopist at the turn of the XIX century). It is interesting to point out that ArthurSchuster found the solution to the Lane–Emden equation with exponent 5, i.e., to theequation,

−∆u = u5,

in R3, with u > 0 going to zero at infinity. The solution is given by

u =31/4

(1 + |x|2)1/2.

(A. Schuster, On the internal constitution of the Sun, Brit. Assoc. Rept. pp. 427–429,1883). Since the Lane–Emden equation for exponent 5 is the Euler–Lagrange equationfor the minimizer of the Sobolev quotient, this is precisely the function that, modulotranslations and dilations, gives the best Sobolev constant. For a nice autobiographyof Arthur Schuster see A. Schuster, Biographical fragments, Mc Millan & Co., London,(1932).

ii) A very nice short biography of Marc Kac was written by H. P. McKean [Mark Kac inBibliographical Memoirs, National Academy of Science, 59, 214–235 (1990); available onthe web (page by page) at http://www.nap.edu/books/0309041988/html/214.html]. Thereader may want to read his own autobiography: Mark Kac, Enigmas of Chance, Harperand Row, NY, 1985 [reprinted in 1987 in paperback by The University of California Press].For his article in the American Mathematical Monthly, op. cit., Mark Kac obtained the1968 Chauvenet Prize of the Mathematical Association of America.

iii) It is interesting to remark that the values of the first Dirichlet eigenvalues of theGWW domains were obtained experimentally by S. Sridhar and A. Kudrolli, Experimentson Not “Hearing the Shape” of Drums, Physical Review Letters, 72, 2175–2178 (1994).In this article one can find the details of the physics experiments performed by theseauthors using very thin electromagnetic resonant cavities with the shape of the Gordon–Webb–Wolpert (GWW) domains. This is the first time that the approximate numericalvalues of the first 25 eigenvalues of the two GWW were obtained. The correspondingeigenfunctions are also displayed. A quick reference to the transplantation method ofPierre Berard is also given in this article, including the transplantation matrix connectingthe two isospectral domains. The reader may want to check the web page of S. Sridhar’sLab (http://sagar.physics.neu.edu/) for further experiments on resonating cavities, theireigenvalues and eigenfunctions, as well as on experiments on quantum chaos.

iv) The numerical computation of the eigenvalues and eigenfunctions of the pair of GWWisospectral domains was obtained by Tobin A. Driscoll, Eigenmodes of isospectral domains,SIAM Review 39, 1–17 (1997).


v) In its simplified form, the Gordon–Webb–Wolpert domains (GWW domains) are made

of seven congruent rectangle isosceles triangles. Certainly the GWW domains have the

same area, perimeter and connectivity. The GWW domains are not convex. Hence, one

may still ask the question whether one can hear the shape of a convex drum. There are

examples of convex isospectral domains in higher dimension (see e.g. C. Gordon and D.

Webb, Isospectral convex domains in Euclidean Spaces, Math. Res. Letts. 1, 539–545

(1994), where they construct convex isospectral domains in Rn, n ≥ 4). Remark: For an

update of the Sunada Method, and its applications see the article of Robert Brooks [The

Sunada Method, in Tel Aviv Topology Conference “Rothenberg Festschrift” 1998, Contem-

prary Mathematics 231, 25–35 (1999); electronically available at: http://www.math.technion.ac.il/ rbrooks]

CHAPTER 3

Rearrangements

3.1. Definition and basic properties

For many problems of functional analysis it is useful to replace some functionby an equimeasurable but more symmetric one. This method, which was first intro-duced by Hardy and Littlewood, is called rearrangement or Schwarz symmetriza-tion [HLP64]. Among several other applications, it plays an important role in theproofs of isoperimetric inequalities like the Rayleigh–Faber–Krahn inequality or thePayne–Polya–Weinberger inequality (see Chapter 4 and Chapter 6 below). In thefollowing we present some basic definitions and theorems concerning sphericallysymmetric rearrangements.

We let Ω be a measurable subset of Rn and write |Ω| for its Lebesgue measure,which may be finite or infinite. If it is finite we write Ω? for an open ball with thesame measure as Ω, otherwise we set Ω? = Rn. We consider a measurable functionu : Ω → R and assume either that |Ω| is finite or that u decays at infinity, i.e.,|x ∈ Ω : |u(x)| > t| is finite for every t > 0.

Definition 3.1. The function

µ(t) = |x ∈ Ω : |u(x)| > t|, t ≥ 0

is called distribution function of u.

From this definition it is straightforward to check that µ(t) is a decreasing (non–increasing), right-continuous function on R+ with µ(0) = |sprt u| and sprt µ =[0, ess sup |u|).

Definition 3.2.• The decreasing rearrangement u] : R+ → R+ of u is the distribution

function of µ.• The symmetric decreasing rearrangement u? : Ω? → R+ of u is defined by

u?(x) = u](Cn|x|n), where Cn = πn/2[Γ(n/2+1)]−1 is the measure of then-dimensional unit ball.

Because µ is a decreasing function, Definition 3.2 implies that u] is an essentiallyinverse function of µ. The names for u] and u? are justified by the following twolemmas:

Lemma 3.3.(a) The function u] is decreasing, u](0) = esssup |u| and sprtu] = [0, |sprtu|)(b) u](s) = min t ≥ 0 : µ(t) ≤ s(c) u](s) =

∫∞0

χ[0,µ(t))(s) dt

(d) |s ≥ 0 : u](s) > t| = |x ∈ Ω : |u(x)| > t| for all t ≥ 0.(e) s ≥ 0 : u](s) > t = [0, µ(t)) for all t ≥ 0.

15

16 3. REARRANGEMENTS

Proof. Part (a) is a direct consequence of the definition of u], keeping in mindthe general properties of distribution functions stated above. The representationformula in part (b) follows from

u](s) = |w ≥ 0 : µ(w) > s| = supw ≥ 0 : µ(w) > s = minw ≥ 0 : µ(w) ≤ s,where we have used the definition of u] in the first step and then the monotonicityand right-continuity of µ. Part (c) is a consequence of the ‘layer-cake formula’, seeTheorem 7.1 in the appendix. To prove part (d) we need to show that

(3.1) s ≥ 0 : u](s) > t = [0, µ(t)).

Indeed, if s is an element of the left hand side of (3.1), then by Lemma 3.3, part(b), we have

minw ≥ 0 : µ(w) ≤ s > t.

But this means that µ(t) > s, i.e., s ∈ [0, µ(t)). On the other hand, if s is anelement of the right hand side of (3.1), then s < µ(t) which implies again by part(b) that

u](s) = minw ≥ 0 : µ(w) ≤ s ≥ minw ≥ 0 : µ(w) < µ(t) > t,

i.e., s is also an element of the left hand side. Finally, part (e) is a direct consequencefrom part (d). ¤

It is straightforward to transfer the statements of Lemma 3.3 to the symmetricdecreasing rearrangement:

Lemma 3.4.(a) The function u? is spherically symmetric and radially decreasing.(b) The measure of the level set x ∈ Ω? : u?(x) > t is the same as the

measure of x ∈ Ω : |u(x)| > t for any t ≥ 0.

From Lemma 3.3 (c) and Lemma 3.4 (b) we see that the three functions u,u] and u? have the same distribution function and therefore they are said to beequimeasurable. Quite analogous to the decreasing rearrangements one can alsodefine increasing ones:

Definition 3.5.• If the measure of Ω is finite, we call u](s) = u](|Ω| − s) the increasing

rearrangement of u.• The symmetric increasing rearrangement u? : Ω? → R+ of u is defined by

u?(x) = u](Cn|x|n)

3.2. Main theorems

Rearrangements are a useful tool of functional analysis because they consider-ably simplify a function without changing certain properties or at least changingthem in a controllable way. The simplest example is the fact that the integral of afunction’s absolute value is invariant under rearrangement. A bit more generally,we have:

Theorem 3.6. Let Φ be a continuous increasing map from R+ to R+ withΦ(0) = 0. Then∫

Ω?

Φ(u?(x)) dx =∫

Ω

Φ(|u(x)|) dx =∫

Ω?

Φ(u?(x)) dx.

3.2. MAIN THEOREMS 17

Proof. The theorem follows directly from Theorem 7.1 in the appendix: If wechoose m( dx) = dx, the right hand side of (7.1) takes the same value for v = |u|,v = u? and v = u?. The conditions on Φ are necessary since Φ(t) = ν([0, t)) musthold for some measure ν on R+. ¤

For later reference we state a rather specialized theorem, which is an estimateon the rearrangement of a spherically symmetric function that is defined on anasymmetric domain:

Theorem 3.7. Assume that uΩ : Ω → R+ is given by uΩ(x) = u(|x|), whereu : R+ → R+ is a non-negative decreasing (resp. increasing) function. Thenu?

Ω(x) ≤ u(|x|) (resp. uΩ?(x) ≥ u(|x|)) for every x ∈ Ω?.

Proof. Assume first that u is a decreasing function. The layer–cake represen-tation for u?

Ω is

u?Ω(x) = u](Cn|x|n) =

∫ ∞

0

χ[0,|x∈Ω:uΩ(x)>t|)(Cn|x|n) dt

≤∫ ∞

0

χ[0,|x∈Rn:u(|x|)>t|)(Cn|x|n) dt

=∫ ∞

0

χx∈Rn:u(|x|)>t(x) dt

= u(|x|)¤

The product of two functions changes in a controllable way under rearrange-ment:

Theorem 3.8. Suppose that u and v are measurable and non-negative functionsdefined on some Ω ⊂ Rn with finite measure. Then

(3.2)∫

R+u](s) v](s) ds ≥

∫

Ω

u(x) v(x) dx ≥∫

R+u](s) v](s) ds

and

(3.3)∫

Ω?

u?(x) v?(x) dx ≥∫

Ω

u(x) v(x) dx ≥∫

Ω?

u?(x) v?(x) dx.

Proof. We first show that for every measurable Ω′ ⊂ Ω and every measurablev : Ω → R+ the relation

(3.4)∫ |Ω′|

0

v](s) ds ≥∫

Ω′v dx ≥

∫ |Ω′|

0

v](s) ds

holds: We can assume without loss of generality that v is integrable. Then thelayer-cake formula (see Theorem 7.1 in the appendix) gives

(3.5) v =∫ ∞

0

χx∈Ω:v(x)>t dt and v] =∫ ∞

0

χ[0,µ(t)) dt.

Hence, ∫

Ω′v dx =

∫ ∞

0

|Ω′ ∩ x ∈ Ω : v(x) > t| dt,

∫ |Ω′|

0

v](s) ds =∫ ∞

0

min(|Ω′|, |x ∈ Ω : v(x) > t|) dt.


The first inequality in (3.4) follows. The second inequality in (3.4) can be estab-lished with the help of the first:

∫ |Ω′|

0

v] ds =∫ |Ω|

0

v] ds−∫ |Ω|

|Ω′|v] ds

=∫

Ω

v dx−∫ |Ω|−|Ω′|

0

v] ds

≤∫

Ω

v dx−∫

Ω\Ω′v dx =

∫

Ω′v dx.

Now assume that u and v are measurable, non-negative and - without loosinggenerality - integrable. Since we can replace v by u in the equations (3.5), we have

∫

Ω

u(x)v(x) dx =∫ ∞

0

dt

∫

x∈Ω:u(x)>t

v(x) dx,

∫ ∞

0

u](s)v](s) ds =∫ ∞

0

dt

∫ µ(t)

0

v](s) ds,

where µ is the distribution function of u. On the other hand, the first inequality in(3.4) tells us that

∫

x∈Ω:u(x)>t

v(x) dx ≤∫ µ(t)

0

v](s) ds

for every non-negative t, such that the first inequality in (3.2) follows. The secondpart of (3.2) can be proven completely analogously, and the inequalities (3.3) are adirect consequence of (3.2). ¤

3.3. Gradient estimates

The integral of a function’s gradient over the boundary of a level set can beestimated in terms of the distribution function:

Theorem 3.9. Assume that u : Rn → R is Lipschitz continuous and decays atinfinity, i.e., the measure of Ωt := x ∈ Rn : |u(x)| > t is finite for every positivet. If µ is the distribution function of u then

(3.6)∫

∂Ωt

|∇u|Hn−1( dx) ≥ −n2C2/nn

µ(t)2−2/n

µ′(t).

Remark: Here Hn(A) denotes the n–dimensional Hausdorff measure of the set A(see, e.g., [Fe69]).

Proof. On the one hand, by the classical isoperimetric inequality we have

(3.7)∫

∂Ωt

Hn−1( dx) ≥ nC1/nn |Ωt|1−1/n = nC1/n

n µ(t)1−1/n.

3.3. GRADIENT ESTIMATES 19

On the other hand, we can use the Cauchy-Schwarz inequality to get∫

∂Ωt

Hn−1( dx) =∫

∂Ωt

√|∇u|√|∇u|Hn−1( dx)

≤(∫

∂Ωt

|∇u|Hn−1( dx))1/2 (∫

∂Ωt

1|∇u|Hn−1( dx)

)1/2

.

The last integral in the above formula can be replaced by −µ′(t) according toFederer’s coarea formula (see, [Fe69]). The result is

(3.8)∫

∂Ωt

Hn−1( dx) ≤(∫

∂Ωt

|∇u|Hn−1( dx))1/2

(−µ′(t))1/2.

Comparing the equations (3.7) and (3.8) yields Theorem 3.9. ¤

Integrals that involve the norm of the gradient can be estimated using thefollowing important theorem:

Theorem 3.10. Let Φ : R+ → R+ be a Young function, i.e., Φ is increasingand convex with Φ(0) = 0. Suppose that u : Rn → R is Lipschitz continuous anddecays at infinity. Then∫

Rn

Φ(|∇u?(x)|) dx ≤∫

Rn

Φ(|∇u(x)|) dx.

For the special case Φ(t) = t2 Theorem 3.10 states that the ‘energy expectationvalue’ of a function decreases under symmetric rearrangement, a fact that is key tothe proof of the Rayleigh–Faber–Krahn inequality (see Section 4.1).

Proof. Theorem 3.10 is a consequence of the following chain of (in)equalities,the second step of which follows from Lemma 3.11 below.

∫

Rn

Φ(|∇u|) dx =∫ ∞

0

dsdds

∫

x∈Rn:|u(x)|>u∗(s)

Φ(|∇u|) dx

≥∫ ∞

0

ds Φ(−nC1/n

n s1−1/n du∗

ds(s)

)

=∫

Rn

Φ(|∇u?|) dx.

¤

Lemma 3.11. Let u and Φ be as in Theorem 3.10. Then for almost everypositive s holds

(3.9)dds

∫


Φ(|∇u|) dx ≥ Φ(−nC1/n

n s1−1/n du∗

ds(s)

).

Proof. First we prove Lemma 3.11 for the special case of Φ being the identity.If s > |sprt u| then (3.9) is clearly true since both sides vanish. Thus we can assumethat 0 < s < |sprt u|. For all 0 ≤ a < b < |sprt u| we show that

(3.10)∫

x∈Rn:u∗(a)>|u(x)|>u∗(b)

|∇u(x)|dx ≥ nC1/nn a1−1/n(u∗(a)− u∗(b)).


The statement (3.10) is proven by the following chain of inequalities, in which wefirst use Federer’s coarea formula, then the classical isoperimetric inequality in Rn

and finally the monotonicity of the integrand:

l.h.s. of (3.10) =∫ u∗(a)

u∗(b)Hn−1x ∈ Rn : |u(x)| = tdt

≥∫ u∗(a)

u∗(b)nC1/n

n |x ∈ Rn : |u(x)| ≥ t|1−1/n dt

≥ nC1/nn |x ∈ Rn : |u(x)| ≥ u∗(a)|1−1/n · (u∗(a)− u∗(b))

≥ r.h.s. of (3.10).

In the case of Φ being the identity, Lemma 3.11 follows from (3.10): Replace b bya + ε with some ε > 0, multiply both sides by ε−1 and then let ε go to zero.

It remains to show that equation (3.9) holds for almost every s > 0 if Φ is notthe identity but some general Young function. From the monotonicity of u∗ followsthat for almost every s > 0 either du∗

ds is zero or there is a neighborhood of s whereu∗ is continuous and decreases strictly. In the first case there is nothing to prove,thus we can assume the second one. Then we have

(3.11) |x ∈ Rn : u∗(s) ≥ |u(x)| > u∗(s + ε)| = ε

for small enough ε > 0. Consequently, we can apply Jensen’s inequality to get

1ε

∫

x∈Rn:u∗(s)≥|u(x)|>u∗(s+h)

Φ(|∇u(x)|) dx ≥ Φ

1

ε

∫

x∈Rn:u∗(s)≥|u(x)|>u∗(s+h)

|∇u(x)| dx

.

Taking the limit ε ↓ 0, this yields

dds

∫


Φ(|∇u(x)|) dx ≥ Φ

d

ds

∫


|∇u(x)|dx

.

Since we have already proven Lemma 3.11 for the case of Φ being the identity, wecan apply it to the argument of Φ on the right hand side of the above inequality.The statement of Lemma 3.11 for general Φ follows. ¤


i) Rearrangements of functions were introduced by G. Hardy and J. E. Littlewood. Theirresults are contained in the classical book, G.H. Hardy, J. E. Littlewood, J.E., and G.Polya, Inequalities, 2d ed., Cambridge University Press, 1952. The fact that the L2 normof the gradient of a function decreases under rearrangements was proven by Faber andKrahn (see references below). A more modern proof as well as many results on rear-rangements and their applications to PDE’s can be found in G. Talenti, Elliptic equationsand rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3, 697–718 (1976).(The reader may want to see also the article by E.H. Lieb, Existence and uniqueness ofthe minimizing solution of Choquard’s nonlinear equation, Studies in Appl. Math. 57,93–105 (1976/77), for an alternative proof of the fact that the L2 norm of the gradient


decreases under rearrangements using heat kernel techniques). An excellent expositoryreview on rearrangements of functions (with a good bibliography) can be found in Tal-enti, G., Inequalities in rearrangement invariant function spaces, in Nonlinear analysis,function spaces and applications, Vol. 5 (Prague, 1994), 177–230, Prometheus, Prague,1994. (available at the website: http://www.emis.de/proceedings/Praha94/). The Rieszrearrangement inequality is the assertion that for nonnegative measurable functions f, g, hin Rn, we have∫

Rn×Rn

f(y)g(x− y)h(x)dx dy ≤∫

Rn×Rn

f?(y)g?(x− y)h?(x)dx dy

. For n = 1 the inequality is due to F. Riesz, Sur une inegalite integrale, Journal of theLondon Mathematical Society 5, 162–168 (1930). For general n is due to S.L. Sobolev, Ona theorem of functional analysis, Mat. Sb. (NS) 4, 471–497 (1938) [the English translationappears in AMS Translations (2) 34, 39–68 (1963)]. The cases of equality in the Rieszinequality were studied by A. Burchard, Cases of equality in the Riesz rearrangementinequality, Annals of Mathematics 143 499–627 (1996) (this paper also has an interestinghistory of the problem).

ii) Rearrangements of functions have been extensively used to prove symmetry propertiesof positive solutions of nonlinear PDE’s. See, e.g., Kawohl, Bernhard, Rearrangementsand convexity of level sets in PDE. Lecture Notes in Mathematics, 1150. Springer-Verlag,Berlin (1985), and references therein.

iii) There are different types of rearrangements of functions. For an interesting approachto rearrangements see, Brock, Friedemann and Solynin, Alexander Yu. An approach tosymmetrization via polarization. Trans. Amer. Math. Soc. 352 1759–1796 (2000). Thisapproach goes back through Baernstein–Taylor (Duke Math. J. 1976), who cite Ahlafors(book on “Conformal invariants”, 1973), who in turn credits Hardy and Littlewood.

CHAPTER 4

The Rayleigh–Faber–Krahn inequality

4.1. The Euclidean case

Many isoperimetric inequalities have been inspired by the question which geo-metrical layout of some physical system maximizes or minimizes a certain quantity.One may ask, for example, how matter of a given mass density must be distributedto minimize its gravitational energy, or which shape a conducting object must haveto maximize its electrostatic capacity. The most famous question of this kind wasput forward at the end of the 19th century by Lord Rayleigh in his work on thetheory of sound [R45]: He conjectured that among all drums of the same area andthe same tension the circular drum produces the lowest fundamental frequency.This statement was proven independently in the 1920s by Faber [F23] and Krahn[K25, K26].

To treat the problem mathematically, we consider an open bounded domainΩ ⊂ R2 which matches the shape of the drum. Then the oscillation frequenciesof the drum are given by the eigenvalues of the Laplace operator −∆Ω

D on Ω withDirichlet boundary conditions, up to a constant that depends on the drum’s tensionand mass density. In the following we will allow the more general case Ω ⊂ Rn forn ≥ 2, although the physical interpretation as a drum only makes sense if n = 2.We define the Laplacian −∆Ω

D via the quadratic–form approach, i.e., it is the uniqueself–adjoint operator in L2(Ω) which is associated with the closed quadratic form

h[Ψ] =∫

Ω

|∇Ψ|2 dx, Ψ ∈ H10 (Ω).

Here H10 (Ω), which is a subset of the Sobolev space W 1,2(Ω), is the closure of

C∞0 (Ω) with respect to the form norm

(4.1) | · |2h = h[·] + || · ||L2(Ω).

For more details about the important question of how to define the Laplace operatoron arbitrary domains and subject to different boundary conditions we refer thereader to [D96, BS87].

The spectrum of −∆ΩD is purely discrete since H1

0 (Ω) is, by Rellich’s theorem,compactly imbedded in L2(Ω) (see, e.g., [BS87]). We write λ1(Ω) for the lowesteigenvalue of −∆Ω

D.

Theorem 4.1 (Rayleigh–Faber–Krahn inequality). Let Ω ⊂ Rn be an openbounded domain with smooth boundary and Ω? ⊂ Rn a ball with the same measureas Ω. Then

λ1(Ω∗) ≤ λ1(Ω)

with equality if and only if Ω itself is a ball.

23

24 4. THE RAYLEIGH–FABER–KRAHN INEQUALITY

Proof. With the powerful mathematical tool of rearrangements (see Chap-ter 3) at hand, the proof of the Rayleigh–Faber–Krahn inequality is actually notdifficult. Let Ψ be the positive normalized first eigenfunction of −∆Ω

D. Since thedomain of a positive self-adjoint operator is a subset of its form domain, we haveΨ ∈ H1

0 (Ω). Then we have Ψ? ∈ H10 (Ω?). Thus we can apply first the min–max

principle and then the Theorems 3.6 and 3.10 to obtain

λ1(Ω?) ≤∫Ω? |∇Ψ?|2 dnx∫Ω? |Ψ∗|2 dnx

≤∫Ω|∇Ψ|2 dnx∫Ω

Ψ2 dnx= λ1(Ω).

¤

The Rayleigh–Faber–Krahn inequality has been extended to a number of differ-ent settings, for example to Laplace operators on curved manifolds or with respectto different measures. In the following we shall give an overview of these general-izations.

4.2. Schrodinger operators

It is not difficult to extend the Rayleigh-Faber-Krahn inequality to Schrodingeroperators, i.e., to operators of the form −∆+V (x). Let Ω ⊂ Rn be an open boundeddomain and V : Rn → R+ a non-negative potential in L1(Ω). Then the quadraticform

hV [u] =∫

Ω

(|∇u|2 + V (x)|u|2) dnx,

defined on

Dom hV = H10 (Ω) ∩

u ∈ L2(Ω) :

∫

Ω

(1 + V (x))|u(x)|2 dnx < ∞

is closed (see, e.g., [D90, D96]). It is associated with the positive self-adjointSchrodinger operator HV = −∆ + V (x). The spectrum of HV is purely discreteand we write λ1(Ω, V ) for its lowest eigenvalue.

Theorem 4.2. Under the assumptions stated above,

λ1(Ω∗, V?) ≤ λ1(Ω, V ).

Proof. Let u1 ∈ Dom hV be the positive normalized first eigenfunction ofHV . Then we have u?

1 ∈ H10 (Ω?) and by Theorem 3.8

∫

Ω?

(1 + V?)u?12 dnx ≤

∫

Ω

(1 + V )u21 dnx < ∞.

Thus u?1 ∈ Dom hV? and we can apply first the min–max principle and then Theo-

rems 3.6, 3.8 and 3.10 to obtain

λ1(Ω?, V?) ≤∫Ω?

(|∇u?1|2 + V?u

?12)

dnx∫Ω? |u?

1|2 dnx

≤∫Ω

(|∇u1|2 + V u21

)dnx∫

Ωu2

1 dnx= λ1(Ω, V ).

¤

4.3. GAUSSIAN SPACE 25

4.3. Gaussian Space

Consider the space Rn (n ≥ 2) endowed with the measure dµ = γ(x) dnx,where

(4.2) γ(x) = (2π)−n/2e−|x|22 ,

is the standard Gaussian density. Since γ(x) is a Gauss function we will call(Rn, dµ) the Gaussian space. For any Lebesgue–measurable Ω ⊂ Rn we definethe Gaussian perimeter of Ω by

Pµ(Ω) = sup∫

Ω

((∇− x) · v(x))γ(x) dx : v ∈ C10 (Ω,Rn), ||v||∞ ≤ 1

.

If ∂Ω is sufficiently well-behaved then

Pµ(Ω) =∫

∂Ω

γ(x) dHn−1,

where Hn−1 is the (n−1)–dimensional Hausdorff measure [Fe69]. It has been shownby Borell that in Gaussian space there is an analog to the classical isoperimetricinequality. Yet the sets that minimize the surface (i.e., the Gaussian perimeter) fora given volume (i.e., Gaussian measure) are not balls, as in Euclidean space, buthalf–spaces [B75]. More precisely:

Theorem 4.3. Let Ω ⊂ Rn be open and measurable. Let further Ω] be thehalf-space ~x ∈ Rn : x1 > a, where a ∈ R is chosen such that µ(Ω) = µ(Ω]). Then

Pµ(Ω) ≥ Pµ(Ω])

with equality only if Ω = Ω] up to a rotation.

Next we define the Laplace operator for domains in Gaussian space. We choosean open domain Ω ⊂ Rn with µ(Ω) < µ(Rn) = 1 and consider the function space

H1(Ω, dµ) =

u ∈ W 1,1loc (Ω) such that (u, |∇u|) ∈ L2(Ω, dµ)× L2(Ω, dµ)

,

endowed with the norm

||u||H1(Ω, dµ) = ||u||L2(Ω, dµ) + ||∇u||L2(Ω, dµ).

We define the quadratic form

h[u] =∫

Ω

|∇u|2 dµ

on the closure of C∞0 (Ω) in H1(Ω, dµ). Since H1 is complete, Dom h is also com-plete under its form norm, which is equal to || · ||H1(Ω, dµ). The quadratic form h istherefore closed and associated with a unique positive self-adjoint operator −∆G.Dom h is embedded compactly in L2(Ω, dµ) and therefore the spectrum of −∆G

is discrete. Its eigenfunctions and eigenvalues are solutions of the boundary valueproblem

(4.3)−

n∑j=1

∂∂xj

(γ(x) ∂

∂xju)

= λγ(x)u in Ω,

u = 0 on ∂Ω.

The analog of the Rayleigh–Faber–Krahn inequality for Gaussian Spaces is thefollowing theorem.


Theorem 4.4. Let λ1(Ω) be the lowest eigenvalue of −∆G on Ω and let Ω′ bea half-space of the same Gaussian measure as Ω. Then

λ1(Ω′) ≤ λ1(Ω).

Equality holds if and only if Ω itself is a half-space.

4.4. Spaces of constant curvature

Differential operators can not only be defined for functions in Euclidean space,but also for the more general case of functions on Riemannian manifolds. It istherefore natural to ask whether the isoperimetric inequalities for the eigenvalues ofthe Laplacian can be generalized to such settings as well. In this section we will stateRayleigh–Faber–Krahn type theorems for the spaces of constant non-zero curvature,i.e., for the sphere and the hyperbolic space. Isoperimetric inequalities for thesecond Laplace eigenvalue in these curved spaces will be discussed in Section 6.7.

To start with, we define the Laplacian in hyperbolic space as a self-adjointoperator by means of the quadratic form approach. We realize Hn as the open unitball B = (x1, . . . , xn) :

∑nj=1 x2

j < 1 endowed with the metric

(4.4) ds2 =4|dx|2

(1− |x|2)2and the volume element

(4.5) dV =2n dnx

(1− |x|2)n,

where | · | denotes the Euclidean norm. Let Ω ⊂ Hn be an open domain and assumethat it is bounded in the sense that Ω does not touch the boundary of B. Thequadratic form of the Laplace operator in hyperbolic space is the closure of

(4.6) h[u] =∫

Ω

gij(∂iu)(∂ju) dV, u ∈ C∞0 (Ω).

It is easy to see that the form (4.6) is indeed closeable: Since Ω does not touchthe boundary of B, the metric coefficients gij are bounded from above on Ω. Theyare also bounded from below by gij ≥ 4. Consequently, the form norms of h andits Euclidean counterpart, which is the right hand side of (4.6) with gij replacedby δij , are equivalent. Since the ‘Euclidean’ form is well known to be closeable, hmust also be closeable.

By standard spectral theory, the closure of h induces an unique positive self-adjoint operator −∆H which we call the Laplace operator in hyperbolic space.Equivalence between corresponding norms in Euclidean and hyperbolic space im-plies that the imbedding Dom h → L2(Ω, dV ) is compact and thus the spectrumof −∆H is discrete. For its lowest eigenvalue the following Rayleigh–Faber–Krahninequality holds.

Theorem 4.5. Let Ω ⊂ Hn be an open bounded domain with smooth boundaryand Ω? ⊂ Hn an open geodesic ball of the same measure. Denote by λ1(Ω) andλ1(Ω?) the lowest eigenvalue of the Dirichlet-Laplace operator on the respectivedomain. Then

λ1(Ω?) ≤ λ1(Ω)

with equality only if Ω itself is a geodesic ball.


The Laplace operator −∆S on a domain which is contained in the unit sphereSn can be defined in a completely analogous fashion to −∆H by just replacing themetric gij in (4.6) by the metric of Sn.

Theorem 4.6. Let Ω ⊂ Sn be an open bounded domain with smooth boundaryand Ω? ⊂ Sn an open geodesic ball of the same measure. Denote by λ1(Ω) andλ1(Ω?) the lowest eigenvalue of the Dirichlet-Laplace operator on the respectivedomain. Then

λ1(Ω?) ≤ λ1(Ω)with equality only if Ω itself is a geodesic ball.

The proofs of the above theorems are similar to the proof for the Euclideancase and will be omitted here. A more general Rayleigh–Faber–Krahn theorem forthe Laplace operator on Riemannian manifolds and its proof can be found in thebook of Chavel [C84].

4.5. Robin Boundary Conditions

Yet another generalization of the Rayleigh–Faber–Krahn inequality holds forthe boundary value problem

(4.7)−

n∑j=1

∂2

∂x2ju = λu in Ω,

∂u∂ν + βu = 0 on ∂Ω,

on a bounded Lipschitz domain Ω ⊂ Rn with the outer unit normal ν and someconstant β > 0. This so–called Robin boundary value problem can be interpreted asa mathematical model for a vibrating membrane whose edge is coupled elasticallyto some fixed frame. The parameter β indicates how tight this binding is andthe eigenvalues of (4.7) correspond the the resonant vibration frequencies of themembrane. They form a sequence 0 < λ1 < λ2 ≤ λ3 ≤ . . . (see, e.g., [M85]).

The Robin problem (4.7) is more complicated than the corresponding Dirichletproblem for several reasons. For example, the very useful property of domainmonotonicity does not hold for the eigenvalues of the Robin–Laplacian. That is,if one enlarges the domain Ω in a certain way, the eigenvalues may go up. It isknown though, that a very weak form of domain monotonicity holds, namely thatλ1(B) ≤ λ1(Ω) if B is ball that contains Ω. Another difficulty of the Robin problem,compared to the Dirichlet case, is that the level sets of the eigenfunctions may touchthe boundary. This makes it impossible, for example, to generalize the proof of theRayleigh–Faber–Krahn inequality in a straightforward way. Nevertheless, such anisoperimetric inequality holds, as proven by Daners:

Theorem 4.7. Let Ω ⊂ Rn (n ≥ 2) be a bounded Lipschitz domain, β > 0 aconstant and λ1(Ω) the lowest eigenvalue of (4.7). Then λ1(Ω?) ≤ λ1(Ω).

For the proof of Theorem 4.7, which is not short, we refer the reader to [D06].


i) The Rayleigh–Faber–Krahn inequality is an isoperimetric inequality concerning thelowest eigenvalue of the Laplacian, with Dirichlet boundary condition, on a boundeddomain in Rn (n ≥ 2). Let 0 < λ1(Ω) < λ2(Ω) ≤ λ3(Ω) ≤ . . . be the Dirichlet eigenvaluesof the Laplacian in Ω ⊂ Rn, i.e.,

−∆u = λu in Ω,


u = 0 on the boundary of Ω.

If n = 2, the Dirichlet eigenvalues are proportional to the square of the eigenfrequencies ofan elastic, homogeneous, vibrating membrane with fixed boundary. The Rayleigh–Faber–Krahn inequality for the membrane (i.e., n = 2) states that

λ1 ≥ πj20,1

A,

where j0,1 = 2.4048 . . . is the first zero of the Bessel function of order zero, and A is thearea of the membrane. Equality is obtained if and only if the membrane is circular. Inother words, among all membranes of given area, the circle has the lowest fundamentalfrequency. This inequality was conjectured by Lord Rayleigh (see, Rayleigh, J.W.S., TheTheory of Sound, second edition, London, 1894/1896, pp. 339–340). In 1918, Courant(see R. Courant, Math. Z. 1, 321–328 (1918)) proved the weaker result that among allmembranes of the same perimeter L the circular one yields the least lowest eigenvalue,i.e.,

λ1 ≥ 4π2j20,1

L2,

with equality if and only if the membrane is circular. Rayleigh’s conjecture was proven in-dependently by Faber [F23] and Krahn [K25]. The corresponding isoperimetric inequalityin dimension n,

λ1(Ω) ≥(

1

|Ω|)2/n

C2/nn jn/2−1,1,

was proven by Krahn [K26]. Here jm,1 is the first positive zero of the Bessel function

Jm, |Ω| is the volume of the domain, and Cn = πn/2/Γ(n/2 + 1) is the volume of then–dimensional unit ball. Equality is attained if and only if Ω is a ball. For more detailssee, R.D. Benguria, Rayleigh–Faber–Krahn Inequality, in Encyclopaedia of Mathematics,Supplement III, Managing Editor: M. Hazewinkel, Kluwer Academic Publishers, pp. 325–327, (2001).

ii) A natural question to ask concerning the Rayleigh–Faber–Krahn inequality is the ques-tion of stability. If the lowest eigenvalue of a domain Ω is within ε (positive and suffi-ciently small) of the isoperimetric value λ1(Ω

∗), how close is the domain Ω to being a ball?The problem of stability for (convex domains) concerning the Rayleigh–Faber–Krahn in-equality was solved by Antonios Melas (Melas, A.D., The stability of some eigenvalueestimates, J. Differential Geom. 36, 19–33 (1992)). In the same reference, Melas alsosolved the analogous stability problem for convex domains with respect to the PPW in-equality (see Chapter 6 below). The work of Melas has been extended to the case of theSzego–Weinberger inequality (for the first nontrivial Neumann eigenvalue) by Xu, Youyu,The first nonzero eigenvalue of Neumann problem on Riemannian manifolds, J. Geom.Anal. 5 151–165 (1995), and to the case of the PPW inequality on speces of constantcurvature by Andres Avila, Stability results for the first eigenvalue of the Laplacian ondomains in space forms, J. Math. Anal. Appl. 267, 760–774 (2002). In this connection itis worth mentioning related results on the isoperimetric inequality of R. Hall, A quantita-tive isoperimetric inequality in n–dimensional space, J. Reine Angew Math. 428 (1992),161–176, as well as recent results of Maggi, Pratelli and Fusco (recently reviewed by F.Maggi in Bull. Amer. Math. Soc. 45 (2008), 367–408.

iii) The analog of the Faber–Krahn inequality for domains in the sphere Sn was proven bySperner, Emanuel, Jr. Zur Symmetrisierung von Funktionen auf Spharen, Math. Z. 134(1973), 317–327


iv) For isoperimetric inequalities for the lowest eigenvalue of the Laplace–Beltrami oper-ator on manifolds, see, e.g., the book by Chavel, Isaac, Eigenvalues in Riemanniangeometry. Pure and Applied Mathematics, 115. Academic Press, Inc., Orlando, FL,1984, (in particular Chapters IV and V), and also the articles, Chavel, I. and Feldman,E. A. Isoperimetric inequalities on curved surfaces. Adv. in Math. 37, 83–98 (1980),and Bandle, Catherine, Konstruktion isoperimetrischer Ungleichungen der mathematis-chen Physik aus solchen der Geometrie, Comment. Math. Helv. 46, 182–213 (1971).

CHAPTER 5

The Szego–Weinberger inequality

In analogy to the Rayleigh–Faber–Krahn inequality for the Dirichlet–Laplacianone may ask which shape of a domain maximizes certain eigenvalues of the Laplaceoperator with Neumann boundary conditions. Of course, this question is trivialfor the lowest Neumann eigenvalue, which is always zero. In 1952 Kornhauser andStakgold [KS52] conjectured that the ball maximizes the first non-zero Neumanneigenvalue among all domains of the same volume. This was first proven in 1954 bySzego [S54] for two-dimensional simply connected domains, using conformal map-pings. Two years later his result was generalized general domains in any dimensionby Weinberger [W56], who came up with a new strategy for the proof.

Although the Szego–Weinberger inequality appears to be the analog for Neu-mann eigenvalues of the Rayleigh–Faber–Krahn inequality, its proof is completelydifferent. The reason is that the first non-trivial Neumann eigenfunction must beorthogonal to the constant function, and thus it must have a change of sign. Thesimple symmetrization procedure that is used to establish the Rayleigh–Faber–Krahn inequality can therefore not work.

In general, when dealing with Neumann problems, one has to take into accountthat the spectrum of the respective Laplace operator on a bounded domain isvery unstable under perturbations. One can change the spectrum arbitrarily muchby only a slight modification of the domain, and if the boundary is not smoothenough, the Laplacian may even have essential spectrum. A sufficient condition forthe spectrum of −∆Ω

N to be purely discrete is that Ω is bounded and has a Lipschitzboundary [D96]. We write 0 = µ0(Ω) < µ1(Ω) ≤ µ2(Ω) ≤ . . . for the sequence ofNeumann eigenvalues on such a domain Ω.

Theorem 5.1 (Szego–Weinberger inequality). Let Ω ⊂ Rn be an open boundeddomain with smooth boundary such that the Laplace operator on Ω with Neumannboundary conditions has purely discrete spectrum. Then

(5.1) µ1(Ω) ≤ µ1(Ω?),

where Ω? ⊂ Rn is a ball with the same n-volume as Ω. Equality holds if and onlyif Ω itself is a ball.

Proof. By a standard separation of variables one shows that µ1(Ω?) is n-folddegenerate and that a basis of the corresponding eigenspace can be written in theform g(r)rjr

−1j=1,...,n. The function g can be chosen to be positive and satisfiesthe differential equation

(5.2) g′′ +n− 1

rg′ +

(µ1(Ω?)− n− 1

r2

)g = 0, 0 < r < r1,

where r1 is the radius of Ω?. Further, g(r) vanishes at r = 0 and its derivative hasits first zero at r = r1. We extend g by defining g(r) = limr′↑r1 g(r′) for r ≥ r1.

31

32 5. THE SZEGO–WEINBERGER INEQUALITY

Then g is differentiable on R and if we set fj(~r) := g(r)rjr−1 then fj ∈ W 1,2(Ω) for

j = 1 . . . , n. To apply the min-max principle with fj as a test function for µ1(Ω)we have to make sure that fj is orthogonal to the first (trivial) eigenfunction, i.e.,that

(5.3)∫

Ω

fj dnr = 0, j = 1, . . . , n.

We argue that this can be achieved by some shift of the domain Ω: Since Ω isbounded we can find a ball B that contains Ω. Now define the vector field ~b : Rn →Rn by its components

bj(~v) =∫

Ω+~v

fj(~r) dnr, ~v ∈ Rn.

For ~v ∈ ∂B we have

~v ·~b(~v) =∫

Ω+~v

~v · ~rr

g(r) dnr

=∫

Ω

~v · (~r + ~v)|~r + ~v| g(|~r + ~v|) dnr

≥∫

Ω

|~v|2 − |v| · |r||~r + ~v| g(|~r + ~v|) dnr > 0.

Thus~b is a vector field that points outwards on every point of ∂B. By an applicationof the Brouwer’s fixed–point theorem (see Theorem 7.3 in the Appendix) this meansthat ~b(~v0) = 0 for some ~v0 ∈ B. Thus, if we shift Ω by this vector, condition (5.3)is satisfied and we can apply the min-max principle with the fj as test functionsfor the first non-zero eigenvalue:

µ1(Ω) ≤∫Ω|∇fj | dnr∫Ω

f2j dnr

=

∫Ω

(g′2(r)r2

j r−2 + g2(r)(1− r2j r−2)r−2

)dnr

∫Ω

g2r2j r−2 dnr

.

We multiply each of these inequalities by the denominator and sum up over j toobtain

(5.4) µ1(Ω) ≤∫Ω

B(r) dnr∫Ω

g2(r) dnr

with B(r) = g′2(r) + (n − 1)g2(r)r−2. Since r1 is the first zero of g′, the functiong is non-decreasing. The derivative of B is

B′ = 2g′g′′ + 2(n− 1)(rgg′ − g2)r−3.

For r ≥ r1 this is clearly negative since g is constant there. For r < r1 we can useequation (5.2) to show that

B′ = −2µ1(Ω?)gg′ − (n− 1)(rg′ − g)2r−3 < 0.

If the following we will use the method of rearrangements, which was described inChapter 3. To avoid confusions, we use a more precise notation at this point: We


introduce BΩ : Ω → R , BΩ(~r) = B(r) and analogously gΩ : Ω → R, gΩ(~r) = g(r).Then equation (5.4) yields, using Theorem 3.7 in the third step:

(5.5) µ1(Ω) ≤∫Ω

BΩ(~r) dnr∫Ω

g2Ω(~r) dnr

=

∫Ω? B?

Ω(~r) dnr∫Ω? g2

?Ω(~r) dnr≤

∫Ω? B(r) dnr∫Ω? g2(r) dnr

= µ1(Ω?)

Equality holds obviously if Ω is a ball. In any other case the third step in (5.5) isa strict inequality. ¤

It is rather straightforward to generalize the Szego–Weinberger inequality todomains in hyperbolic space. For domains on spheres, on the other hand, thecorresponding inequality has not been established yet in full generality. At present,the most general result is due to Ashbaugh and Benguria: In [AB95] they show thatan analog of the Szego–Weinberger inequality holds for domains that are containedin a hemisphere.


i) In 1952, Kornhauser and Stakgold [Journal of Mathematics and Physics 31, 45–54(1952)] conjectured that the lowest nontrivial Neumann eigenvalue for a smooth boundeddomain Ω in R2 satisfies the isoperimetric inequality

µ1(Ω) ≤ µ1(Ω∗) =

πp2

A,

where Ω∗ is a disk with the same area as Ω, and p = 1.8412 . . . is the first positive zero ofthe derivative of the Bessel function J1. This conjecture was proven by G. Szego in 1954,using conformal maps [see, G. Szego, Inequalities for certain eigenvalues of a membraneof given area, J. Rational Mech. Anal. 3, 343–356 (1954)]. The extension to n dimensionswas proven by H. Weinberger [H. F. Weinberger, J. Rational Mech. Anal. 5, 633–636(1956)].

ii) For the case of mixed boundary conditions, Marie–Helene Bossel [Membranes elastiquementliees inhomogenes ou sur une surface: une nouvelle extension du theoreme isoperimetriquede Rayleigh–Faber–Krahn, Z. Angew. Math. Phys. 39, 733–742 (1988)] proved the analogof the Rayleigh–Faber–Krahn inequality.

iii) Very recently, A. Girouard, N. Nadirashvili and I. Polterovich proved that the secondpositive eigenvalue of a bounded simply connected planar domain of a given area doesnot exceed the first positive Neumann eigenvalue on a disk of a twice smaller area (see,Maximization of the second positive Neumann eigenvalue for planar domains, preprint(2008)). For a review of optimization of eigenvalues with respect to the geometry of thedomain, see the recent monograph of A. Henrot [H06].

CHAPTER 6

The Payne–Polya–Weinberger inequality

6.1. Introduction

A further isoperimetric inequality is concerned with the second eigenvalue ofthe Dirichlet–Laplacian on bounded domains. In 1955 Payne, Polya and Wein-berger (PPW) showed that for any open bounded domain Ω ⊂ R2 the boundλ2(Ω)/λ1(Ω) ≤ 3 holds [PPW55, PPW56]. Based on exact calculations for sim-ple domains they also conjectured that the ratio λ2(Ω)/λ1(Ω) is maximized whenΩ is a circular disk, i.e., that

(6.1)λ2(Ω)λ1(Ω)

≤ λ2(Ω?)λ1(Ω?)

=j21,1

j20,1

≈ 2.539 for Ω ⊂ R2.

Here, jn,m denotes the mth positive zero of the Bessel function Jn(x). This con-jecture and the corresponding inequalities in n dimensions were proven in 1991 byAshbaugh and Benguria [AB91, AB92a, AB92b]. Since the Dirichlet eigenval-ues on a ball are inversely proportional to the square of the ball’s radius, the ratioλ2(Ω?)/λ1(Ω?) does not depend on the size of Ω?. Thus we can state the PPWinequality in the following form:

Theorem 6.1 (Payne–Polya–Weinberger inequality). Let Ω ⊂ Rn be an openbounded domain and S1 ⊂ Rn a ball such that λ1(Ω) = λ1(S1). Then

(6.2) λ2(Ω) ≤ λ2(S1)

with equality if and only if Ω is a ball.

Here the subscript 1 on S1 reflects the fact that the ball S1 has the same firstDirichlet eigenvalue as the original domain Ω The inequalities (6.1) and (6.2) areequivalent in Euclidean space in view of the mentioned scaling properties of theeigenvalues. Yet when one considers possible extensions of the PPW inequality toother settings, where λ2/λ1 varies with the radius of the ball, it turns out that anestimate in the form of Theorem 6.1 is the more natural result. In the case of adomain on a hemisphere, for example, λ2/λ1 on balls is an increasing function ofthe radius. But by the Rayleigh–Faber–Krahn inequality for spheres the radius ofS1 is smaller than the one of the spherical rearrangement Ω?. This means that anestimate in the form of Theorem 6.1, interpreted as

λ2(Ω)λ1(Ω)

≤ λ2(S1)λ1(S1)

, Ω, S1 ⊂ Sn,

is stronger than an inequality of the type (6.1).On the other hand, we will see that in the hyperbolic space λ2/λ1 on balls is

a strictly decreasing function of the radius. In this case we can apply the follow-ing argument to see that an estimate of the type (6.1) cannot possibly hold true:

35

36 6. THE PAYNE–POLYA–WEINBERGER INEQUALITY

Consider a domain Ω that is constructed by attaching very long and thin tentaclesto the ball B. Then the first and second eigenvalues of the Laplacian on Ω arearbitrarily close to the ones on B. The spherical rearrangement of Ω though canbe considerably larger than B. This means that

λ2(Ω)λ1(Ω)

≈ λ2(B)λ1(B)

>λ2(Ω?)λ1(Ω?)

, B, Ω ⊂ Hn,

clearly ruling out any inequality in the form of (6.1).The proof of the PPW inequality (6.2) is somewhat similar to that of the

Szego–Weinberger inequality (see Chapter 5), but considerably more difficult. Theadditional complications mainly stem from the fact that in the Dirichlet case thefirst eigenfunction of the Laplacian is not known explicitly, while in the Neumanncase it is just constant. We will give the full proof of the PPW inequality in thefollowing three sections. Since it is quite long, a brief outline is in order:

The proof is organized in six steps. In the first one we use the min–max principleto derive an estimate for the eigenvalue gap λ2(Ω) − λ1(Ω), depending on a testfunction for the second eigenvalue. In the second step we define such a function andthen show in the third step that it actually satisfies all requirements to be used inthe gap formula. In the fourth step we put the test function into the gap inequalityand then estimate the result with the help of rearrangement techniques. Thesedepend on the monotonicity properties of two functions g and B, which are to bedefined in the proof, and on a Chiti comparison argument. The later is a specialcomparison result which establishes a crossing property between the symmetricdecreasing rearrangement of the first eigenfunction on Ω and the first eigenfunctionon S1. We end up with the inequality λ2(Ω)−λ1(Ω) ≤ λ2(S1)−λ1(S1), which yields(6.2). In the remaining two steps we prove the mentioned monotonicity propertiesand the Chiti comparison result. We remark that from the Rayleigh–Faber–Krahninequality follows S1 ⊂ Ω?, a fact that is used in the proof of the Chiti comparisonresult. Although it enters in a rather subtle manner, the Rayleigh–Faber–Krahninequality is an important ingredient of the proof of the PPW inequality.

6.2. Proof of the Payne–Polya–Weinberger inequality

First step: We derive the ‘gap formula’ for the first two eigenvalues of theDirichlet–Laplacian on Ω. We call u1 : Ω → R+ the positive normalized firsteigenfunction of −∆D

Ω . To estimate the second eigenvalue we will use the testfunction Pu1, where P : Ω → R is is chosen such that Pu1 is in the form domainof −∆D

Ω and

(6.3)∫

Ω

Pu21 drn = 0.

Then we conclude from the min–max principle that

λ2(Ω)− λ1(Ω) ≤∫Ω

(|∇(Pu1)|2 − λ1P2u2

1

)drn

∫Ω

P 2u21 drn

=

∫Ω

(|∇P |2u21 + (∇P 2)u1∇u1 + P 2|∇u1|2 − λ1P

2u21

)drn

∫Ω

P 2u21 drn

(6.4)

If we perform an integration by parts on the second summand in the numeratorof (6.4), we see that all summands except the first cancel. We obtain the gap

6.2. PROOF OF THE PAYNE–POLYA–WEINBERGER INEQUALITY 37

inequality

(6.5) λ2(Ω)− λ1(Ω) ≤∫Ω|∇P |2u2

1 drn

∫Ω

P 2u21 drn

.

Second step: We need to fix the test function P . Our choice will be dictatedby the requirement that equality should hold in (6.5) if Ω is a ball, i.e., if Ω = S1

up to translations. We assume that S1 is centered at the origin of our coordinatesystem and call R1 its radius. We write z1(r) for the first eigenfunction of theDirichlet Laplacian on S1. This function is spherically symmetric with respectto the origin and we can take it to be positive and normalized in L2(S1). Thesecond eigenvalue of −∆D

S1in n dimensions is n–fold degenerate and a basis of the

corresponding eigenspace can be written in the form z2(r)rjr−1 with z2 ≥ 0 and

j = 1, . . . , n. This is the motivation to choose not only one test function P , butrather n functions Pj with j = 1, . . . , n. We set

Pj = rjr−1g(r)

with

g(r) =

z2(r)z1(r)

for r < R1,

limr′↑R1z2(r

′)z1(r′)

for r ≥ R1.

We note that Pju1 is a second eigenfunction of −∆DΩ if Ω is a ball which is centered

at the origin.Third step: It is necessary to verify that the Pju1 are admissible test functions.

First, we have to make sure that condition (6.3) is satisfied. We note that Pj

changes when Ω (and u1 with it) is shifted in Rn. Since these shifts do not changeλ1(Ω) and λ2(Ω), it is sufficient to show that Ω can be moved in Rn such that (6.3)is satisfied for all j ∈ 1, . . . , n. To this end we define the function

~b(~v) =∫

Ω+~v

u21(|~r − ~v|)~r

rg(r) drn for ~v ∈ Rn.

Since Ω is a bounded domain, we can choose some closed ball D, centered at theorigin, such that Ω ⊂ D. Then for every ~v ∈ ∂D we have

~v ·~b(~v) =∫

Ω

~v · u21(r)

~r + ~v

|~r + ~v|g(|~r + ~v|) drn

>

∫

Ω

u21(r)

|~v|2 − |~v| · |~r||~r + ~v| g(|~r + ~v|) drn > 0

Thus the continuous vector-valued function ~b(~v) points strictly outwards every-where on ∂D. By Theorem 7.3, which is a consequence of the Brouwer fixed–pointtheorem, there is some ~v0 ∈ D such that ~b(~v0) = 0. Now we shift Ω by this vector,i.e., we replace Ω by Ω−~v0 and u1 by the first eigenfunction of the shifted domain.Then the test functions Pju1 satisfy the condition (6.3).

The second requirement on Pju1 is that it must be in the form domain of−∆D

Ω , i.e., in H10 (Ω): Since u1 ∈ H1

0 (Ω) there is a sequence vn ∈ C1(Ω)n∈Nof functions with compact support such that | · |h − limn→∞ vn = u1, using thedefinition (4.1) of | · |h. The functions Pjvn also have compact support and one cancheck that Pjvn ∈ C1(Ω) (Pj is continuously differentiable since g′(R1) = 0). Wehave | · |h − limn→∞ Pjvn = Pju1 and thus Pju1 ∈ H1

0 (Ω).


Fourth step: We multiply the gap inequality (6.5) by∫

P 2u21 dx and put in our

special choice of Pj to obtain

(λ2 − λ1)∫

Ω

r2j

r2g2(r)u2

1(r) drn ≤∫

Ω

∣∣∣∇(rj

rg(r)

)∣∣∣2

u21(r) drn

=∫

Ω

(∣∣∣∇rj

r

∣∣∣2

g2(r) +r2j

r2g′(r)2

)u2

1(r) drn.

Now we sum these inequalities up over j = 1, . . . , n and then divide again by theintegral on the left hand side to get

(6.6) λ2(Ω)− λ1(Ω) ≤∫Ω

B(r)u21(r) drn

∫Ω

g2(r)u21(r) drn

with

(6.7) B(r) = g′(r)2 + (n− 1)r−2g(r)2.

If the following we will use the method of rearrangements, which was described inChapter 3. To avoid confusions, we use a more precise notation at this point: Weintroduce BΩ : Ω → R , BΩ(~r) = B(r) and analogously gΩ : Ω → R, gΩ(~r) = g(r).Then equation (6.6) can be written as

(6.8) λ2(Ω)− λ1(Ω) ≤∫Ω

BΩ(~r)u21(~r) drn

∫Ω

g2Ω(~r)u2

1(~r) drn.

Then by Theorem 3.8 the following inequality is also true:

(6.9) λ2(Ω)− λ1(Ω) ≤∫Ω? B?

Ω(~r)u?1(~r)

2 drn

∫Ω? g2

Ω?(~r)u?1(~r)2 drn

.

Next we use the very important fact that g(r) is an increasing function and B(r) isa decreasing function, which we will prove in step five below. These monotonicityproperties imply by Theorem 3.7 that B?

Ω(~r) ≤ B(r) and gΩ?(~r) ≥ g(r). Therefore

(6.10) λ2(Ω)− λ1(Ω) ≤∫Ω? B(r)u?

1(r)2 drn

∫Ω? g2(r)u?

1(r)2 drn.

Finally we use the following version of Chiti’s comparison theorem to estimate theright hand side of (6.10):

Lemma 6.2 (Chiti comparison result). There is some r0 ∈ (0, R1) such that

z1(r) ≥ u?1(r) for r ∈ (0, r0) and

z1(r) ≤ u?1(r) for r ∈ (r0, R1).

We remind the reader that the function z1 denotes the first Dirichlet eigenfunc-tion for the Laplacian defined on S1. Applying Lemma 6.2, which will be provenbelow in step six, to (6.10) yields

(6.11) λ2(Ω)− λ1(Ω) ≤∫Ω? B(r)z1(r)2 drn

∫Ω? g2(r)z1(r)2 drn

= λ2(S1)− λ1(S1).

Since S1 was chosen such that λ1(Ω) = λ1(S1) the above relation proves thatλ2(Ω) ≤ λ2(S1). It remains the question: When does equality hold in (6.2)? It isobvious that equality does hold if Ω is a ball, since then Ω = S1 up to translations.On the other hand, if Ω is not a ball, then (for example) the step from (6.10) to(6.11) is not sharp. Thus (6.2) is a strict inequality if Ω is not a ball.

6.3. MONOTONICITY OF B AND g 39

6.3. Monotonicity of B and g

Fifth step: We prove that g(r) is an increasing function and B(r) is a decreasingfunction. In this step we abbreviate λi = λi(S1). The functions z1 and z2 aresolutions of the differential equations

−z′′1 −n− 1

rz′1 − λ1z1 = 0,(6.12)

−z′′2 −n− 1

rz′2 +

(n− 1

r2− λ2

)z2 = 0

with the boundary conditions

(6.13) z′1(0) = 0, z1(R1) = 0, z2(0) = 0, z2(R1) = 0.

We define the function

(6.14) q(r) :=

rg′(r)g(r) for r ∈ (0, R1),

limr′↓0 q(r′) for r = 0,limr′↑R1 q(r′) for r = R1.

Proving the monotonicity of B and g is thus reduced to showing that 0 ≤ q(r) ≤ 1and q′(r) ≤ 0 for r ∈ [0, R1]. Using the definition of g and the equations (6.12),one can show that q(r) is a solution of the Riccati differential equation

(6.15) q′ = (λ1 − λ2)r +(1− q)(q + n− 1)

r− 2q

z′1z1

.

It is straightforward to establish the boundary behavior

q(0) = 1, q′(0) = 0, q′′(0) =2n

((1 +

2n

)λ1 − λ2

)

and

q(R1) = 0.

Lemma 6.3. For 0 ≤ r ≤ R1 we have q(r) ≥ 0.

Proof. Assume the contrary. Then there exist two points 0 < s1 < s2 ≤ R1

such that q(s1) = q(s2) = 0 but q′(s1) ≤ 0 and q′(s2) ≥ 0. If s2 < R1 then theRiccati equation (6.15) yields

0 ≥ q′(s1) = (λ1 − λ2)s1 +n− 1

s1> (λ1 − λ2)s2 +

n− 1s2

= q′(s2) ≥ 0,

which is a contradiction. If s2 = R1 then we get a contradiction in a similar wayby

0 ≥ q′(s1) = (λ1 − λ2)s1 +n− 1

s1> (λ1 − λ2)R1 +

n− 1R1

= 3q′(R1) ≥ 0.

¤

In the following we will analyze the behavior of q′ according to (6.15), consid-ering r and q as two independent variables. For the sake of a compact notation we


will make use of the following abbreviations:

p(r) = z′1(r)/z1(r)Ny = y2 − n + 1

Qy = 2yλ1 + (λ2 − λ1)Nyy−1 − 2(λ2 − λ1)

My = N2y /(2y)− (n− 2)2y/2

We further define the function

(6.16) T (r, y) := −2p(r)y − (n− 2)y + Ny

r− (λ2 − λ1)r.

Then we can write (6.15) as

q′(r) = T (r, q(r)).

The definition of T (r, y) allows us to analyze the Riccati equation for q′ consideringr and q(r) as independent variables. For r going to zero, p is O(r) and thus

T (r, y) =1r

((n− 1 + y)(1− y)) +O(r) for y fixed.

Consequently,

limr→0 T (r, y) = +∞ for 0 ≤ y < 1 fixed,limr→0 T (r, y) = 0 for y = 1 andlimr→0 T (r, y) = −∞ for y > 1 fixed.

The partial derivative of T (r, y) with respect to r is given by

(6.17) T ′ =∂

∂rT (r, y) = −2yp′ +

(n− 2)yr2

+Ny

r2− (λ2 − λ1).

In the points (r, y) where T (r, y) = 0 we have, by (6.16),

(6.18) p|T=0 = −n− 22r

− Ny

2yr− (λ2 − λ1)r

2y.

From (6.12) we get the Riccati equation

(6.19) p′ + p2 +n− 1

rp + λ1 = 0.

Putting (6.18) into (6.19) and the result into (6.17) yields

(6.20) T ′|T=0 =My

r2+

(λ2 − λ1)2

2yr2 + Qy.

Lemma 6.4. There is some r0 > 0 such that q(r) ≤ 1 for all r ∈ (0, r0) andq(r0) < 1.

Proof. Suppose the contrary, i.e., q(r) first increases away from r = 0. Then,because q(0) = 1 and q(R1) = 0 and because q is continuous and differentiable, wecan find two points s1 < s2 such that q := q(s1) = q(s2) > 1 and q′(s1) > 0 > q′(s2).Even more, we can chose s1 and s2 such that q is arbitrarily close to one. Writingq = 1 + ε with ε > 0, we can calculate from the definition of Qy that

Q1+ε = Q1 + εn (λ2 − (1− 2/n)λ1) +O(ε2).

The term in brackets can be estimated by

λ2 − (1− 2/n)λ1 > λ2 − λ1 > 0.

6.4. THE CHITI COMPARISON RESULT 41

We can also assume that Q1 ≥ 0, because otherwise q′′(0) = 2n2 Q1 < 0 and Lemma

6.4 is immediately true. Thus, choosing R1 and r2 such that ε is sufficiently small,we can make sure that Qq > 0.

Now consider T (r, q) as a function of r for our fixed q. We have T (s1, q) > 0 >T (s2, q) and the boundary behavior T (0, q) = −∞. Consequently, T (r, q) changesits sign at least twice on [0, R1] and thus we can find two zeros 0 < s1 < s2 < R1

of T (r, q) such that

(6.21) T ′(s1, q) ≥ 0 and T ′(s2, q) ≤ 0.

But from (6.20), together with Qq > 0, one can see easily that this is impossible,because the right hand side of (6.20) is either positive or increasing (depending onMq). This is a contradiction to our assumption that q first increases away fromr = 0, proving Lemma 6.4. ¤

Lemma 6.5. For all 0 ≤ r ≤ R1 the inequality q′(r) ≤ 0 holds.

Proof. Assume the contrary. Then, because of q(0) = 1 and q(R1) = 0, thereare three points s1 < s2 < s3 in (0, R1) with 0 < q := q(s1) = q(s2) = q(s3) < 1 andq′(s1) < 0, q′(s2) > 0, q′(s3) < 0. Consider the function T (r, q), which coincideswith q′(r) at s1, s2, s3. Taking into account its boundary behavior at r = 0, it isclear that T (r, q) must have at least the sign changes positive-negative-positive-negative. Thus T (r, q) has at least three zeros s1 < s2 < s3 with the properties

T ′(s1, q) ≤ 0, T ′(s2, q) ≥ 0, T ′(s3, q) ≤ 0.

Again one can see from (6.20) that this is impossible, because the term on theright hand side is either a strictly convex or a strictly increasing function of r. Weconclude that Lemma 6.5 is true. ¤

Altogether we have shown that 0 ≤ q(r) ≤ 1 and q′(r) ≤ 0 for all r ∈ (0, R1),which proves that g is increasing and B is decreasing.

6.4. The Chiti comparison result

Sixth step: We prove Lemma 6.2: Here and in the sequel we write short-handλ1 = λ1(Ω) = λ1(S1). We introduce a change of variables via s = Cnrn, whereCn is the volume of the n–dimensional unit ball. Then by Definition 3.2 we haveu]

1(s) = u?1(r) and z]

1(s) = z1(r).

Lemma 6.6. For the functions u]1(s) and z]

1(s) we have

− du]1

ds≤ λ1n

−2C−2/nn sn/2−2

∫ s

0

u]1(w) dw,(6.22)

− dz]1

ds= λ1n

−2C−2/nn sn/2−2

∫ s

0

z]1(w) dw.(6.23)

Proof. We integrate both sides of −∆u1 = λ1u1 over the level set Ωt := ~r ∈Ω : u1(~r) > t and use Gauss’ Divergence Theorem to obtain

(6.24)∫

∂Ωt

|∇u1|Hn−1( dr) =∫

Ωt

λ1 u1(~r) dnr,


where ∂Ωt = ~r ∈ Ω : u1(~r) = t. Now we define the distribution function µ(t) =|Ωt|. Then by Theorem 3.9 we have

(6.25)∫

∂Ωt

|∇u1|Hn−1( dr) ≥ −n2C2/nn

µ(t)2−2/n

µ′(t).

The left sides of (6.24) and (6.25) are the same, thus

−n2C2/nn

µ(t)2−2/n

µ′(t)≤

∫

Ωt

λ1 u1(~r) dnr

=∫ (µ(t)/Cn)1/n

0

nCnrn−1λ1u?1(r) dr.

Now we perform the change of variables r → s on the right hand side of the abovechain of inequalities. We also chose t to be u]

1(s). Using the fact that u]1 and

µ are essentially inverse functions to one another, this means that µ(t) = s andµ′(t)−1 = (u]

1)′(s). The result is (6.22). Equation (6.23) is proven analogously,

with equality in each step. ¤

Lemma 6.6 enables us to prove Lemma 6.2. The function z]1 is continuous on

(0, |S1|) and u]1 is continuous on (0, |Ω?|). By the normalization of u]

1 and z]1 and

because S1 ⊂ Ω? it is clear that either z]1 ≥ u]

1 on (0, |S1|) or u]1 and z]

1 have atleast one intersection on this interval. In the first case there is nothing to prove,simply setting r0 = R1 in Lemma 6.2. In the second case we have to show thatthere is no intersection of u]

1 and z]1 such that u]

1 is greater than z]1 on the left and

smaller on the right. So we assume the contrary, i.e., that there are two points0 ≤ s1 < s2 < |S1| such that u]

1(s) > z]1(s) for s ∈ (s1, s2), u]

1(s2) = z]1(s2) and

either u]1(s1) = z]

1(s1) or s1 = 0. We set

(6.26) v](s) =

u]1(s) on [0, s1] if

∫ s1

0u]

1(s) ds >∫ s1

0z]1(s) ds,

z]1(s) on [0, s1] if

∫ s1

0u]

1(s) ds ≤ ∫ s1

0z]1(s) ds,

u]1(s) on [s1, s2],

z]1(s) on [s2, |S1|].

Then one can convince oneself that because of (6.22) and (6.23)

(6.27) − dv]

ds≤ λ1n

−2C−2/nn sn/2−2

∫ s

0

v](s′) ds′

for all s ∈ [0, |S1|]. Now define the test function v(r) = v](Cnrn). Using theRayleigh-Ritz characterization of λ1, then (6.27) and finally an integration by parts,

6.5. SCHRODINGER OPERATORS 43

we get (if z1 and u1 are not identical)

λ1

∫

S1

v2(r) dx <

∫

S1

|∇v|2 dx =∫ |S1|

0

(nCnrn−1 v]′(s)

)2 ds

≤ −∫ |S1|

0

v]′(s)λ1

∫ s

0

v](s′) ds′ ds

= λ1

∫ |S1|

0

v](s)2 ds− λ1

[v](s)

∫ s

0

v](s′) ds′]S1

0

≤ λ1

∫

S1

v2(r) dx

Comparing the first and the last term in the above chain of (in)equalities reveals acontradiction to our assumption that the intersection point s2 exists, thus provingLemma 6.2.

6.5. Schrodinger operators

Theorem 6.1 can be extended in several directions. One generalization, whichhas been considered by Benguria and Linde in [BL06], is to replace the Laplaceoperator on the domain Ω ⊂ Rn by a Schrodinger operator H = −∆ + V . In thiscase the question arises which is the most suitable comparison operator for H. Inanalogy to the PPW inequality for the Laplacian, it seems natural to compare theeigenvalues of H to those of another Schrodinger operator H = −∆ + V , which isdefined on a ball and has the same lowest eigenvalue as H. The potential V shouldbe spherically symmetric and it should reflect some properties of V , but it will alsohave to satisfy certain requirements in order for the PPW type estimate to hold.The precise result is stated in Theorem 6.7 below, which can be considered as anatural generalization of Theorem 6.1 to Schrodinger operators.

We assume that Ω is open and bounded and that V : Ω → R+ is a non-negativepotential from L1(Ω). Then we can define the Schrodinger operator HV = −∆+Von Ω in the same way as we did in Section 4.2, i.e., HV is positive and self-adjointin L2(Ω) and has purely discrete spectrum. We call λi(Ω, V ) its i-th eigenvalueand, as usual, we write V? for the symmetric increasing rearrangement of V .

Theorem 6.7. Let S1 ⊂ Rn be a ball centered at the origin and of radius R1

and let V : S1 → R+ be a radially symmetric non-negative potential such thatV (r) ≤ V?(r) for all 0 ≤ r ≤ R1 and λ1(Ω, V ) = λ1(S1, V ). If V (r) satisfies theconditions

a) V (0) = V ′(0) = 0 andb) V ′(r) exists and is increasing and convex,

then

(6.28) λ2(Ω, V ) ≤ λ2(S1, V ).

If V is such that V? itself satisfies the conditions a) and b) of the theorem, thebest bound is obtained by choosing V = V? and then adjusting the size of S1 suchthat λ1(Ω, V ) = λ1(S1, V?) holds. (Note that S1 ⊂ Ω? by Theorem 4.2). In thiscase Theorem 6.7 is a typical PPW result and optimal in the sense that equalityholds in (6.28) if Ω is a ball and V = V?. For a general potential V we still get anon-trivial bound on λ2(Ω, V ) though it is not sharp anymore.


For further reference we state the following theorem, which is a direct conse-quence of Theorem 6.7 and Theorem 3.7:

Theorem 6.8. Let V : Rn → R+ be a radially symmetric positive potential thatsatisfies the conditions a) and b) of Theorem 6.1. Further, assume that Ω ⊂ Rn isan open bounded domain and that S1 ⊂ Rn be the open ball (centered at the origin)such that λ1(Ω, V ) = λ1(S1, V ). Then

λ2(Ω, V ) ≤ λ2(S1, V ).

The proof of Theorem 6.7 is similar to the one of Theorem 6.1 and can be foundin [BL06]. One of the main differences occurs in step five (see Section 6.3), sincethe potential V (r) now appears in the Riccati equation for p. It turns out thatthe conditions a) and b) in Theorem 6.7 are required to establish the monotonicityproperties of q. A second important difference is that a second eigenfunction ofa Schrodinger operator with a spherically symmetric potential can not necessarilybe written in the form u2(r)rjr

−1. It has been shown by Ashbaugh and Benguria[AB88] that it can be written in this form if rV (r) is convex. On the other hand,the second eigenfunction is radially symmetric (with a spherical nodal surface) ifrV (r) is concave. This fact, which is also known as the Baumgartner–Grosse–Martin Inequality [BGM84], is another reason why the conditions a) and b) ofTheorem 6.7 are needed.

6.6. Gaussian space

Theorem 6.8 has direct consequences for the eigenvalues of the Laplace operator−∆G in Gaussian space, which had been defined in Section 4.3. In this section wewrite λ−i (Ω) for the i-th eigenvalue of −∆G on some domain Ω.

Theorem 6.9. Let Ω ⊂ Rn be an open bounded domain and assume that S1 ⊂Rn is a ball, centered at the origin, such that λ−1 (Ω) = λ−1 (S1). Then

λ−2 (Ω) ≤ λ−2 (S1).

Proof. If Ψ is an eigenfunction of −∆G on Ω then Ψe−r2/2 is an eigenfunctionof the Dirichlet-Schrodinger operator −∆+r2 on Ω, and vice versa. The eigenvaluesare related by

λ−i (Ω) = λ(Ω, r2 − n),

where we keep using the notation from Section 6.5. Theorem 6.9 now followsdirectly from Theorem 6.8, setting V (r) = r2. ¤

6.7. Spaces of constant curvature

There are generalizations of the Payne-Polya-Weinberger inequality to spaces ofconstant curvature. Ashbaugh and Benguria showed in [AB01] that Theorem 6.1remains valid if one replaces the Euclidean space Rn by a hemisphere of Sn and ‘ball’by ‘geodesic ball’. Similar to the Szego–Weinberger inequality, it is still an openproblem to prove a Payne–Polya–Weinberger result for the whole sphere. Althoughthere seem to be no counterexamples known that rule out such a generalization, theoriginal scheme of proving the PPW inequality is not likely to work. One reasonis that numerical studies show the function g to be not monotone on the wholesphere.

6.7. SPACES OF CONSTANT CURVATURE 45

For the hyperbolic space, on the other hand, things are settled. Following thegeneral lines of the original proof, Benguria and Linde established in [BL07] aPPW type inequality that holds in any space of constant negative curvature.

CHAPTER 7

Appendix

7.1. The layer-cake formula

Theorem 7.1. Let ν be a measure on the Borel sets of R+ such that Φ(t) :=ν([0, t)) is finite for every t > 0. Let further (Ω, Σ,m) be a measure space and v anon-negative measurable function on Ω. Then

(7.1)∫

Ω

Φ(v(x))m( dx) =∫ ∞

0

m(x ∈ Ω : v(x) > t)ν( dt).

In particular, if m is the Dirac measure at some point x ∈ Rn and ν( dt) = dt then(7.1) takes the form

(7.2) v(x) =∫ ∞

0

χy∈Ω:v(y)>t(x) dt.

Proof. Since m(x ∈ Ω : v(x) > t) =∫Ω

χv>t(x)m( dx) we have, usingFubini’s theorem,

∫ ∞

0

m(x ∈ Ω : v(x) > t)ν( dt) =∫

Ω

(∫ ∞

0

χv>t(x)ν( dt))

m( dx).

Theorem 7.1 follows from observing that∫ ∞

0

χv>t(x)ν( dt) =∫ v(x)

0

ν( dt) = Φ(v(x)).

¤

7.2. A consequence of the Brouwer fixed-point theorem

Theorem 7.2 (Brouwer’s fixed-point theorem). Let B ⊂ Rn be the unit ballfor n ≥ 0. If f : B → B is continuous then f has a fixed point, i.e., there is somex ∈ B such that f(x) = x.

The proof appears in many books on topology, e.g., in [M75]. Brouwer’stheorem can be applied to establish the following result:

Theorem 7.3. Let B ⊂ Rn (n ≥ 2) be a closed ball and ~b(~r) a continuous mapfrom B to Rn. If ~b points strictly outwards at every point of ∂B, i.e., if ~b(~r) ·~r > 0for every ~r ∈ ∂B, then ~b has a zero in B.

Proof. Without losing generality we can assume that B is the unit ball cen-tered at the origin. Since ~b is continuous and ~b(~r) · ~r > 0 on ∂B, there are twoconstants 0 < r0 < 1 and p > 0 such that ~b(~r) · ~r > p for every ~r with r0 < |~r| ≤ 1.We show that there is a constant c > 0 such that

| − c~b(~r) + ~r| < 1

47

48 7. APPENDIX

for all ~r ∈ B: In fact, for all ~r with |~r| ≤ r0 the constant c can be any positivenumber below (sup~r∈B |~b(~r)|)−1(1 − r0). The supremum exists because |~b| is con-tinuously defined on a compact set and therefore bounded. On the other hand, forall ~r ∈ B with |~r| > r0 we have

| − c~b(~r) + ~r|2 = c2|~b(~r)|2 − 2c~b(~r) · ~r + |~r|2≤ c2 sup

~r∈B|~b|2 − 2cp + 1,

which is also smaller than one if one chooses c > 0 sufficiently small. Now set

~g(~r) = −c~b(~r) + ~r for ~r ∈ B.

Then ~g is a continuous mapping from B to B and by Theorem 7.2 it has some fixedpoint ~r1 ∈ B, i.e., ~g(~r1) = ~r1 and ~b(~r1) = 0. ¤

Bibliography

[AB88] M.S. Ashbaugh, R.D. Benguria: Log-concavity of the ground state of Schrodinger oper-ators: A new proof of the Baumgartner-Grosse-Martin inequality, Physical Letters A 131,273–276 (1988).

[AB91] M. S. Ashbaugh and R. D. Benguria, Proof of the Payne–Polya–Weinberger conjecture,Bull. Amer. Math. Soc. 25, 19–29 (1991).

[AB92a] M. S. Ashbaugh and R. D. Benguria, A sharp bound for the ratio of the first two eigen-values of Dirichlet Laplacians and extensions, Annals of Math. 135, 601–628 (1992).

[AB92b] M. S. Ashbaugh and R. D. Benguria, A second proof of the Payne–Polya–Weinbergerconjecture, Commun. Math. Phys. 147, 181–190 (1992).

[AB95] M. S. Ashbaugh and R. D. Benguria, Sharp Upper Bound to the First Nonzero NeumannEigenvalue for Bounded Domains in Spaces of Constant Curvature, Journal of the LondonMathematical Society (2) 52, 402–416 (1995).

[AB01] M. S. Ashbaugh and R. D. Benguria, A Sharp Bound for the Ratio of the First TwoDirichlet Eigenvalues of a Domain in a Hemisphere of Sn, Transactions of the AmericanMathematical Society 353, 1055–1087 (2001).

[B80] C. Bandle: Isoperimetric Inequalities and Applications, Pitman Monographs and Studiesin Mathematics, vol. 7, Pitman, Boston (1980).

[BGM84] B. Baumgartner, H. Grosse, A. Martin: The Laplacian of the potential and the orderof energy levels, Physics Letters 146B, 363–366 (1984).

[BL06] R.D. Benguria, H. Linde: A second eigenvalue bound for the Dirichlet Schrodinger oper-ator, Commun. Math. Phys. 267, 741–755 (2006).

[BL07] R.D. Benguria, H. Linde: A second eigenvalue bound for the Dirichlet Laplacian in hy-perbolic space, Duke Mathematical Journal 140, 245–279 (2007).

[Be92] Pierre Berard, Transplantation et isospectralite I, Math. Ann. 292, 547–559 (1992).[Be93] Pierre Berard, Transplantation et isospectralite II, J. London Math. Soc. 48, 565-576

(1993).[Be93b] Pierre Berard, Domaines plans isospectraux a la Gordon–Web–Wolpert: une preuve ele-

mentaire, Afrika Math. 1, 135–146 (1993).[BS87] M. S. Birman, M. Z. Solomjak: Spectral theory of self-adjoint operators in Hilbert Space,

D. Reidel Publishing Company, Dordrecht (1987).[B75] C. Borell: The Brunn-Minkowski inequality in Gauss space, Invent. Math. 30, 207–211

(1975).[Br88] Robert Brooks, Constructing Isospectral Manifolds, Amer. Math. Monthly 95, 823–839

(1988).[B92] P. Buser: Geometry and spectra of compact Riemannian surfaces, Progress in Mathematics

106, Birkhauser, Boston (1992).[C84] I. Chavel: Eigenvalues in Riemannian geometry, Academic Press, Inc., NY (1984).[CH53] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 1, Interscience Pub-

lishers, New York (1953).[D06] D. Daners: A Faber-Krahn inequality for Robin problems in any space dimension, Mathe-

matische Annalen 335, 767–785 (2006).[D90] E.B. Davies: Heat kernels and spectral theory, paperback edition, Cambridge University

Press, Cambridge, UK (1990).[D96] E.B. Davies: Spectral theory and differential operators, Cambridge University Press, Cam-

bridge, UK (1996).

49

50 BIBLIOGRAPHY

[F23] G. Faber: Beweis, dass unter allen homogenen Membranen von gleicher Flache und gleicherSpannung die kreisformige den tiefsten Grundton gibt, Sitzungberichte der mathematisch-physikalischen Klasse der Bayerischen Akademie der Wissenschaften zu Munchen Jahrgang,pp. 169–172 (1923).

[Fe69] H. Federer, Geometric Measure Theory, Springer Verlag, New York (1969).[GWW92] C. Gordon, D. Webb, and S. Wolpert, Isospectral plane domains and surfaces via

Riemannian orbifolds, Invent. Math. 110, 1–22 (1992).[H06] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Collection Frontiers

in Mathematics, Birkhauser, Basel (2006).[HLP64] G.H. Hardy, J.E. Littlewood and G. Polya: Inequalities, Cambridge Univ. Press, Cam-

bridge, UK (1964).[K66] Mark Kac, Can one hear the shape of a drum?, American Mathematical Monthly 73, 1–23

(1966).[KS52] E. T. Kornhauser, I. Stakgold: A variational theorem for ∇2u + λu = 0 and its applica-

tions, J. Math. and Physics 31, 45–54 (1952).

[K25] E. Krahn: Uber eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann.94, 97–100 (1925).

[K26] E. Krahn: Uber Minimaleigenschaften der Kugel in drei und mehr Dimensionen, ActaComm. Univ. Tartu (Dorpat) A9, 1–44 (1926). [English translation: Minimal properties of thesphere in three and more dimensions, Edgar Krahn 1894–1961: A Centenary Volume,U. Lumiste and J. Peetre, editors, IOS Press, Amsterdam, The Netherlands, pp. 139–174(1994).]

[LL97] E. H. Lieb, M. Loss: Analysis, Graduate Studies in Mathematics Vol. 14, American Math-ematical Society, Providence, RI (1997).

[McH94] K. P. McHale, Eigenvalues of the Laplacian, “Can you Hear the Shape of a Drum?”,Master’s Project, Mathematics Department, University of Missouri, Columbia, MO (1994).

[McKS67] H.P. McKean and I.M. Singer, Curvature and the eigenvalues of the Laplacian, Journalof Differential Geometry 1, 662–670 (1967).

[M85] V. G. Maz’ja: Sobolev spaces, Springer Series in Soviet Mathematics. Springer-Verlag,Berlin (1985). Translated from the Russian by T.O. Shaposhnikova.

[Mi64] John Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad.Sc. 51, 542 (1964).

[M75] J.R. Munkres: Topology, A first course, Prentice-Hall, Englewood Cliffs (1975).[O80] R. Osserman, Isoperimetric inequalities and eigenvalues of the Laplacian. in Proceedings

of the International Congress of Mathematicians (Helsinki, 1978), pp. 435–442,Acad. Sci. Fennica, Helsinki, (1980).

[P67] L.E. Payne, Isoperimetric inequalities and their applications, SIAM Review 9, 453–488(1967).

[PPW55] L. E. Payne, G. Polya, and H. F. Weinberger, Sur le quotient de deux frequences propresconsecutives, Comptes Rendus Acad. Sci. Paris 241, 917–919 (1955).

[PPW56] L. E. Payne, G. Polya, and H. F. Weinberger, On the ratio of consecutive eigenvalues,J. Math. and Phys. 35, 289–298 (1956).

[Pj54] A. Pleijel, A study of certain Green’s functions with applications in the theory of vibratingmembranes, Arkiv for Mathematik, 2, 553–569 (1954).

[PSz51] G. Polya and G. Szego, Isoperimetric Inequalities in Mathematical Physics, PrincetonUniversity Press, Princeton, NJ (1951).

[R45] J. W. S. Rayleigh: The Theory of Sound, 2nd. ed. revised and enlarged (in 2 vols.), DoverPublications, New York, (1945) (republication of the 1894/1896 edition).

[RSI] M. Reed, B. Simon: Mathematical Physics, Vol. I, Academic Press Inc., NY (1980).[RSIII] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. 3, Academic

Press, NY (1979).[RSIV] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 4: Anal-

ysis of Operators, Academic Press, NY (1978).

[Ri859] B. Riemann, Uber die Anzahl der Primzahlen unter einer gegebenen Grosse, Monats-berichte der Berliner Akademie, pp. 671–680 (1859).

[SrKu94] S. Sridhar and A. Kudrolli, Experiments on Not “Hearing the Shape” of Drums, PhysicalReview Letters 72, 2175–2178 (1994).

BIBLIOGRAPHY 51

[Su85] T. Sunada, Riemannian Coverings and Isospectral Manifolds, Annals of Math. 121, 169–186 (1985).

[S54] G. Szego: Inequalities for certain eigenvalues of a membrane of given area, J. RationalMech. Anal. 3, 343–356 (1954).

[T76] G. Talenti: Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa (4) 3,697–718 (1976).

[W56] H. F. Weinberger: An isoperimetric inequality for the n-dimensional free membrane prob-lem, J. Rational Mech. Anal. 5, 633–636 (1956).

[We11] H. Weyl, Uber die asymptotische Verteilung der Eigenwerte, Nachr. Akad. Wiss. GottingenMath.–Phys., Kl. II 110–117 (1911).

[We50] H. Weyl, Ramifications, old and new, of the eigenvalue problem Bull. Am. Math. Soc. 56,115–139 (1950).